Proofgold Signed Transaction

Definition False^False := ∀ x0 : ο . x0
Definition not^not := λ x0 : ο . x0 ⟶ False
Definition 8b6ad.. := λ x0 : ι → ι → ο . λ x1 x2 x3 x4 . ∀ x5 : ο . ((x1 = x2 ⟶ ∀ x6 : ο . x6) ⟶ (x1 = x3 ⟶ ∀ x6 : ο . x6) ⟶ (x2 = x3 ⟶ ∀ x6 : ο . x6) ⟶ (x1 = x4 ⟶ ∀ x6 : ο . x6) ⟶ (x2 = x4 ⟶ ∀ x6 : ο . x6) ⟶ (x3 = x4 ⟶ ∀ x6 : ο . x6) ⟶ not (x0 x1 x2) ⟶ not (x0 x1 x3) ⟶ not (x0 x2 x3) ⟶ not (x0 x1 x4) ⟶ not (x0 x2 x4) ⟶ not (x0 x3 x4) ⟶ x5) ⟶ x5
Definition c5756.. := λ x0 : ι → ι → ο . λ x1 x2 x3 x4 x5 . ∀ x6 : ο . (8b6ad.. x0 x1 x2 x3 x4 ⟶ (x1 = x5 ⟶ ∀ x7 : ο . x7) ⟶ (x2 = x5 ⟶ ∀ x7 : ο . x7) ⟶ (x3 = x5 ⟶ ∀ x7 : ο . x7) ⟶ (x4 = x5 ⟶ ∀ x7 : ο . x7) ⟶ not (x0 x1 x5) ⟶ not (x0 x2 x5) ⟶ x0 x3 x5 ⟶ x0 x4 x5 ⟶ x6) ⟶ x6
Definition f831d.. := λ x0 : ι → ι → ο . λ x1 x2 x3 x4 x5 x6 . ∀ x7 : ο . (c5756.. x0 x1 x2 x3 x4 x5 ⟶ (x1 = x6 ⟶ ∀ x8 : ο . x8) ⟶ (x2 = x6 ⟶ ∀ x8 : ο . x8) ⟶ (x3 = x6 ⟶ ∀ x8 : ο . x8) ⟶ (x4 = x6 ⟶ ∀ x8 : ο . x8) ⟶ (x5 = x6 ⟶ ∀ x8 : ο . x8) ⟶ x0 x1 x6 ⟶ x0 x2 x6 ⟶ x0 x3 x6 ⟶ x0 x4 x6 ⟶ not (x0 x5 x6) ⟶ x7) ⟶ x7
Definition 18ba2.. := λ x0 : ι → ι → ο . λ x1 x2 x3 x4 x5 x6 x7 . ∀ x8 : ο . (f831d.. x0 x1 x2 x3 x4 x5 x6 ⟶ (x1 = x7 ⟶ ∀ x9 : ο . x9) ⟶ (x2 = x7 ⟶ ∀ x9 : ο . x9) ⟶ (x3 = x7 ⟶ ∀ x9 : ο . x9) ⟶ (x4 = x7 ⟶ ∀ x9 : ο . x9) ⟶ (x5 = x7 ⟶ ∀ x9 : ο . x9) ⟶ (x6 = x7 ⟶ ∀ x9 : ο . x9) ⟶ x0 x1 x7 ⟶ x0 x2 x7 ⟶ x0 x3 x7 ⟶ x0 x4 x7 ⟶ not (x0 x5 x7) ⟶ not (x0 x6 x7) ⟶ x8) ⟶ x8
Known neq_i_sym^neq_i_sym : ∀ x0 x1 . (x0 = x1 ⟶ ∀ x2 : ο . x2) ⟶ x1 = x0 ⟶ ∀ x2 : ο . x2
Theorem 40e1f.. : ∀ x0 . ∀ x1 : ι → ι → ο . (∀ x2 . x2 ∈ x0 ⟶ ∀ x3 . x3 ∈ x0 ⟶ x1 x2 x3 ⟶ x1 x3 x2) ⟶ ∀ x2 . x2 ∈ x0 ⟶ ∀ x3 . x3 ∈ x0 ⟶ ∀ x4 . x4 ∈ x0 ⟶ ∀ x5 . x5 ∈ x0 ⟶ ∀ x6 . x6 ∈ x0 ⟶ ∀ x7 . x7 ∈ x0 ⟶ ∀ x8 . x8 ∈ x0 ⟶ 18ba2.. x1 x2 x3 x4 x5 x6 x7 x8 ⟶ 18ba2.. x1 x2 x3 x4 x5 x6 x8 x7 (proof)
Definition 2f869.. := λ x0 : ι → ι → ο . λ x1 x2 x3 x4 . ∀ x5 : ο . ((x1 = x2 ⟶ ∀ x6 : ο . x6) ⟶ (x1 = x3 ⟶ ∀ x6 : ο . x6) ⟶ (x2 = x3 ⟶ ∀ x6 : ο . x6) ⟶ (x1 = x4 ⟶ ∀ x6 : ο . x6) ⟶ (x2 = x4 ⟶ ∀ x6 : ο . x6) ⟶ (x3 = x4 ⟶ ∀ x6 : ο . x6) ⟶ not (x0 x1 x2) ⟶ not (x0 x1 x3) ⟶ not (x0 x2 x3) ⟶ not (x0 x1 x4) ⟶ not (x0 x2 x4) ⟶ x0 x3 x4 ⟶ x5) ⟶ x5
Definition 5a3b5.. := λ x0 : ι → ι → ο . λ x1 x2 x3 x4 x5 . ∀ x6 : ο . (2f869.. x0 x1 x2 x3 x4 ⟶ (x1 = x5 ⟶ ∀ x7 : ο . x7) ⟶ (x2 = x5 ⟶ ∀ x7 : ο . x7) ⟶ (x3 = x5 ⟶ ∀ x7 : ο . x7) ⟶ (x4 = x5 ⟶ ∀ x7 : ο . x7) ⟶ not (x0 x1 x5) ⟶ x0 x2 x5 ⟶ not (x0 x3 x5) ⟶ not (x0 x4 x5) ⟶ x6) ⟶ x6
Definition 455db.. := λ x0 : ι → ι → ο . λ x1 x2 x3 x4 x5 x6 . ∀ x7 : ο . (5a3b5.. x0 x1 x2 x3 x4 x5 ⟶ (x1 = x6 ⟶ ∀ x8 : ο . x8) ⟶ (x2 = x6 ⟶ ∀ x8 : ο . x8) ⟶ (x3 = x6 ⟶ ∀ x8 : ο . x8) ⟶ (x4 = x6 ⟶ ∀ x8 : ο . x8) ⟶ (x5 = x6 ⟶ ∀ x8 : ο . x8) ⟶ x0 x1 x6 ⟶ not (x0 x2 x6) ⟶ not (x0 x3 x6) ⟶ x0 x4 x6 ⟶ x0 x5 x6 ⟶ x7) ⟶ x7
Definition 70d65.. := λ x0 : ι → ι → ο . λ x1 x2 x3 x4 x5 x6 x7 . ∀ x8 : ο . (455db.. x0 x1 x2 x3 x4 x5 x6 ⟶ (x1 = x7 ⟶ ∀ x9 : ο . x9) ⟶ (x2 = x7 ⟶ ∀ x9 : ο . x9) ⟶ (x3 = x7 ⟶ ∀ x9 : ο . x9) ⟶ (x4 = x7 ⟶ ∀ x9 : ο . x9) ⟶ (x5 = x7 ⟶ ∀ x9 : ο . x9) ⟶ (x6 = x7 ⟶ ∀ x9 : ο . x9) ⟶ x0 x1 x7 ⟶ x0 x2 x7 ⟶ x0 x3 x7 ⟶ not (x0 x4 x7) ⟶ not (x0 x5 x7) ⟶ not (x0 x6 x7) ⟶ x8) ⟶ x8
Theorem 3ad8d.. : ∀ x0 . ∀ x1 : ι → ι → ο . (∀ x2 . x2 ∈ x0 ⟶ ∀ x3 . x3 ∈ x0 ⟶ x1 x2 x3 ⟶ x1 x3 x2) ⟶ ∀ x2 . x2 ∈ x0 ⟶ ∀ x3 . x3 ∈ x0 ⟶ ∀ x4 . x4 ∈ x0 ⟶ ∀ x5 . x5 ∈ x0 ⟶ ∀ x6 . x6 ∈ x0 ⟶ ∀ x7 . x7 ∈ x0 ⟶ ∀ x8 . x8 ∈ x0 ⟶ 70d65.. x1 x2 x3 x4 x5 x6 x7 x8 ⟶ 70d65.. x1 x2 x5 x6 x3 x4 x8 x7 (proof)
Definition 180f5.. := λ x0 : ι → ι → ο . λ x1 x2 x3 x4 . ∀ x5 : ο . ((x1 = x2 ⟶ ∀ x6 : ο . x6) ⟶ (x1 = x3 ⟶ ∀ x6 : ο . x6) ⟶ (x2 = x3 ⟶ ∀ x6 : ο . x6) ⟶ (x1 = x4 ⟶ ∀ x6 : ο . x6) ⟶ (x2 = x4 ⟶ ∀ x6 : ο . x6) ⟶ (x3 = x4 ⟶ ∀ x6 : ο . x6) ⟶ not (x0 x1 x2) ⟶ not (x0 x1 x3) ⟶ not (x0 x2 x3) ⟶ not (x0 x1 x4) ⟶ x0 x2 x4 ⟶ x0 x3 x4 ⟶ x5) ⟶ x5
Definition 45422.. := λ x0 : ι → ι → ο . λ x1 x2 x3 x4 x5 . ∀ x6 : ο . (180f5.. x0 x1 x2 x3 x4 ⟶ (x1 = x5 ⟶ ∀ x7 : ο . x7) ⟶ (x2 = x5 ⟶ ∀ x7 : ο . x7) ⟶ (x3 = x5 ⟶ ∀ x7 : ο . x7) ⟶ (x4 = x5 ⟶ ∀ x7 : ο . x7) ⟶ not (x0 x1 x5) ⟶ x0 x2 x5 ⟶ x0 x3 x5 ⟶ not (x0 x4 x5) ⟶ x6) ⟶ x6
Definition 85e71.. := λ x0 : ι → ι → ο . λ x1 x2 x3 x4 x5 x6 . ∀ x7 : ο . (45422.. x0 x1 x2 x3 x4 x5 ⟶ (x1 = x6 ⟶ ∀ x8 : ο . x8) ⟶ (x2 = x6 ⟶ ∀ x8 : ο . x8) ⟶ (x3 = x6 ⟶ ∀ x8 : ο . x8) ⟶ (x4 = x6 ⟶ ∀ x8 : ο . x8) ⟶ (x5 = x6 ⟶ ∀ x8 : ο . x8) ⟶ x0 x1 x6 ⟶ not (x0 x2 x6) ⟶ not (x0 x3 x6) ⟶ x0 x4 x6 ⟶ x0 x5 x6 ⟶ x7) ⟶ x7
Definition 843b8.. := λ x0 : ι → ι → ο . λ x1 x2 x3 x4 x5 x6 x7 . ∀ x8 : ο . (85e71.. x0 x1 x2 x3 x4 x5 x6 ⟶ (x1 = x7 ⟶ ∀ x9 : ο . x9) ⟶ (x2 = x7 ⟶ ∀ x9 : ο . x9) ⟶ (x3 = x7 ⟶ ∀ x9 : ο . x9) ⟶ (x4 = x7 ⟶ ∀ x9 : ο . x9) ⟶ (x5 = x7 ⟶ ∀ x9 : ο . x9) ⟶ (x6 = x7 ⟶ ∀ x9 : ο . x9) ⟶ x0 x1 x7 ⟶ x0 x2 x7 ⟶ x0 x3 x7 ⟶ not (x0 x4 x7) ⟶ not (x0 x5 x7) ⟶ not (x0 x6 x7) ⟶ x8) ⟶ x8
Theorem 75481.. : ∀ x0 . ∀ x1 : ι → ι → ο . (∀ x2 . x2 ∈ x0 ⟶ ∀ x3 . x3 ∈ x0 ⟶ x1 x2 x3 ⟶ x1 x3 x2) ⟶ ∀ x2 . x2 ∈ x0 ⟶ ∀ x3 . x3 ∈ x0 ⟶ ∀ x4 . x4 ∈ x0 ⟶ ∀ x5 . x5 ∈ x0 ⟶ ∀ x6 . x6 ∈ x0 ⟶ ∀ x7 . x7 ∈ x0 ⟶ ∀ x8 . x8 ∈ x0 ⟶ 843b8.. x1 x2 x3 x4 x5 x6 x7 x8 ⟶ 843b8.. x1 x2 x5 x6 x3 x4 x8 x7 (proof)
Theorem 122f9.. : ∀ x0 . ∀ x1 : ι → ι → ο . (∀ x2 . x2 ∈ x0 ⟶ ∀ x3 . x3 ∈ x0 ⟶ x1 x2 x3 ⟶ x1 x3 x2) ⟶ ∀ x2 . x2 ∈ x0 ⟶ ∀ x3 . x3 ∈ x0 ⟶ ∀ x4 . x4 ∈ x0 ⟶ ∀ x5 . x5 ∈ x0 ⟶ ∀ x6 . x6 ∈ x0 ⟶ ∀ x7 . x7 ∈ x0 ⟶ ∀ x8 . x8 ∈ x0 ⟶ 70d65.. x1 x2 x3 x4 x5 x6 x7 x8 ⟶ 70d65.. x1 x2 x6 x5 x4 x3 x8 x7 (proof)
Theorem e8292.. : ∀ x0 . ∀ x1 : ι → ι → ο . (∀ x2 . x2 ∈ x0 ⟶ ∀ x3 . x3 ∈ x0 ⟶ x1 x2 x3 ⟶ x1 x3 x2) ⟶ ∀ x2 . x2 ∈ x0 ⟶ ∀ x3 . x3 ∈ x0 ⟶ ∀ x4 . x4 ∈ x0 ⟶ ∀ x5 . x5 ∈ x0 ⟶ ∀ x6 . x6 ∈ x0 ⟶ ∀ x7 . x7 ∈ x0 ⟶ ∀ x8 . x8 ∈ x0 ⟶ 843b8.. x1 x2 x3 x4 x5 x6 x7 x8 ⟶ 843b8.. x1 x2 x6 x5 x4 x3 x8 x7 (proof)
Param 4402e.. : ι → (ι → ι → ο) → ο
Param cf2df.. : ι → (ι → ι → ο) → ο
Definition Subq^Subq := λ x0 x1 . ∀ x2 . x2 ∈ x0 ⟶ x2 ∈ x1
Param setminus^setminus : ι → ι → ι
Param Sing^Sing : ι → ι
Definition 2b028.. := λ x0 : ι → ι → ο . λ x1 x2 x3 x4 x5 . ∀ x6 : ο . (8b6ad.. x0 x1 x2 x3 x4 ⟶ (x1 = x5 ⟶ ∀ x7 : ο . x7) ⟶ (x2 = x5 ⟶ ∀ x7 : ο . x7) ⟶ (x3 = x5 ⟶ ∀ x7 : ο . x7) ⟶ (x4 = x5 ⟶ ∀ x7 : ο . x7) ⟶ not (x0 x1 x5) ⟶ x0 x2 x5 ⟶ x0 x3 x5 ⟶ x0 x4 x5 ⟶ x6) ⟶ x6
Definition 9ab39.. := λ x0 : ι → ι → ο . λ x1 x2 x3 x4 x5 x6 . ∀ x7 : ο . (2b028.. x0 x1 x2 x3 x4 x5 ⟶ (x1 = x6 ⟶ ∀ x8 : ο . x8) ⟶ (x2 = x6 ⟶ ∀ x8 : ο . x8) ⟶ (x3 = x6 ⟶ ∀ x8 : ο . x8) ⟶ (x4 = x6 ⟶ ∀ x8 : ο . x8) ⟶ (x5 = x6 ⟶ ∀ x8 : ο . x8) ⟶ not (x0 x1 x6) ⟶ x0 x2 x6 ⟶ x0 x3 x6 ⟶ x0 x4 x6 ⟶ not (x0 x5 x6) ⟶ x7) ⟶ x7
Definition 8b2a4.. := λ x0 : ι → ι → ο . λ x1 x2 x3 x4 x5 x6 x7 . ∀ x8 : ο . (9ab39.. x0 x1 x2 x3 x4 x5 x6 ⟶ (x1 = x7 ⟶ ∀ x9 : ο . x9) ⟶ (x2 = x7 ⟶ ∀ x9 : ο . x9) ⟶ (x3 = x7 ⟶ ∀ x9 : ο . x9) ⟶ (x4 = x7 ⟶ ∀ x9 : ο . x9) ⟶ (x5 = x7 ⟶ ∀ x9 : ο . x9) ⟶ (x6 = x7 ⟶ ∀ x9 : ο . x9) ⟶ not (x0 x1 x7) ⟶ x0 x2 x7 ⟶ x0 x3 x7 ⟶ x0 x4 x7 ⟶ not (x0 x5 x7) ⟶ not (x0 x6 x7) ⟶ x8) ⟶ x8
Definition and^and := λ x0 x1 : ο . ∀ x2 : ο . (x0 ⟶ x1 ⟶ x2) ⟶ x2
Definition nIn^nIn := λ x0 x1 . not (x0 ∈ x1)
Known setminusE^setminusE : ∀ x0 x1 x2 . x2 ∈ setminus x0 x1 ⟶ and (x2 ∈ x0) (nIn x2 x1)
Definition or^or := λ x0 x1 : ο . ∀ x2 : ο . (x0 ⟶ x2) ⟶ (x1 ⟶ x2) ⟶ x2
Known xm^xm : ∀ x0 : ο . or x0 (not x0)
Known FalseE^FalseE : False ⟶ ∀ x0 : ο . x0
Known 53a3c.. : ∀ x0 . ∀ x1 : ι → ι → ο . (∀ x2 . x2 ∈ x0 ⟶ ∀ x3 . x3 ∈ x0 ⟶ not (x1 x2 x3) ⟶ not (x1 x3 x2)) ⟶ cf2df.. x0 x1 ⟶ ∀ x2 . x2 ∈ x0 ⟶ ∀ x3 . x3 ∈ x0 ⟶ ∀ x4 . x4 ∈ x0 ⟶ ∀ x5 . x5 ∈ x0 ⟶ ∀ x6 . x6 ∈ x0 ⟶ (x2 = x3 ⟶ ∀ x7 : ο . x7) ⟶ (x2 = x4 ⟶ ∀ x7 : ο . x7) ⟶ (x3 = x4 ⟶ ∀ x7 : ο . x7) ⟶ (x2 = x5 ⟶ ∀ x7 : ο . x7) ⟶ (x3 = x5 ⟶ ∀ x7 : ο . x7) ⟶ (x4 = x5 ⟶ ∀ x7 : ο . x7) ⟶ (x2 = x6 ⟶ ∀ x7 : ο . x7) ⟶ (x3 = x6 ⟶ ∀ x7 : ο . x7) ⟶ (x4 = x6 ⟶ ∀ x7 : ο . x7) ⟶ (x5 = x6 ⟶ ∀ x7 : ο . x7) ⟶ not (x1 x2 x3) ⟶ not (x1 x2 x4) ⟶ not (x1 x3 x4) ⟶ not (x1 x2 x5) ⟶ not (x1 x3 x5) ⟶ not (x1 x4 x5) ⟶ not (x1 x2 x6) ⟶ not (x1 x3 x6) ⟶ not (x1 x4 x6) ⟶ not (x1 x5 x6) ⟶ False
Known 61345.. : ∀ x0 . ∀ x1 : ι → ι → ο . (∀ x2 . x2 ∈ x0 ⟶ ∀ x3 . x3 ∈ x0 ⟶ x1 x2 x3 ⟶ x1 x3 x2) ⟶ 4402e.. x0 x1 ⟶ ∀ x2 . x2 ∈ x0 ⟶ ∀ x3 . x3 ∈ x0 ⟶ ∀ x4 . x4 ∈ x0 ⟶ (x2 = x3 ⟶ ∀ x5 : ο . x5) ⟶ (x2 = x4 ⟶ ∀ x5 : ο . x5) ⟶ (x3 = x4 ⟶ ∀ x5 : ο . x5) ⟶ x1 x2 x3 ⟶ x1 x2 x4 ⟶ x1 x3 x4 ⟶ False
Known Subq_tra^Subq_tra : ∀ x0 x1 x2 . x0 ⊆ x1 ⟶ x1 ⊆ x2 ⟶ x0 ⊆ x2
Known setminus_Subq^{setminus_Subq} : ∀ x0 x1 . setminus x0 x1 ⊆ x0
Known SingI^SingI : ∀ x0 . x0 ∈ Sing x0
Theorem b1a55.. : ∀ x0 x1 . ∀ x2 : ι → ι → ο . (∀ x3 . x3 ∈ x1 ⟶ ∀ x4 . x4 ∈ x1 ⟶ x2 x3 x4 ⟶ x2 x4 x3) ⟶ 4402e.. x1 x2 ⟶ cf2df.. x1 x2 ⟶ ∀ x3 . x3 ∈ x1 ⟶ x0 ⊆ setminus x1 (Sing x3) ⟶ ∀ x4 . x4 ∈ x0 ⟶ ∀ x5 . x5 ∈ x0 ⟶ ∀ x6 . x6 ∈ x0 ⟶ ∀ x7 . x7 ∈ x0 ⟶ ∀ x8 . x8 ∈ x0 ⟶ ∀ x9 . x9 ∈ x0 ⟶ ∀ x10 . x10 ∈ x0 ⟶ 8b2a4.. x2 x4 x5 x6 x7 x8 x9 x10 ⟶ ∀ x11 : ο . (x2 x4 x3 ⟶ not (x2 x5 x3) ⟶ not (x2 x6 x3) ⟶ not (x2 x7 x3) ⟶ not (x2 x8 x3) ⟶ not (x2 x9 x3) ⟶ not (x2 x10 x3) ⟶ x11) ⟶ (x2 x4 x3 ⟶ x2 x5 x3 ⟶ not (x2 x6 x3) ⟶ not (x2 x7 x3) ⟶ not (x2 x8 x3) ⟶ not (x2 x9 x3) ⟶ not (x2 x10 x3) ⟶ x11) ⟶ (x2 x4 x3 ⟶ not (x2 x5 x3) ⟶ x2 x6 x3 ⟶ not (x2 x7 x3) ⟶ not (x2 x8 x3) ⟶ not (x2 x9 x3) ⟶ not (x2 x10 x3) ⟶ x11) ⟶ (x2 x4 x3 ⟶ x2 x5 x3 ⟶ x2 x6 x3 ⟶ not (x2 x7 x3) ⟶ not (x2 x8 x3) ⟶ not (x2 x9 x3) ⟶ not (x2 x10 x3) ⟶ x11) ⟶ (x2 x4 x3 ⟶ not (x2 x5 x3) ⟶ not (x2 x6 x3) ⟶ x2 x7 x3 ⟶ not (x2 x8 x3) ⟶ not (x2 x9 x3) ⟶ not (x2 x10 x3) ⟶ x11) ⟶ (x2 x4 x3 ⟶ x2 x5 x3 ⟶ not (x2 x6 x3) ⟶ x2 x7 x3 ⟶ not (x2 x8 x3) ⟶ not (x2 x9 x3) ⟶ not (x2 x10 x3) ⟶ x11) ⟶ (x2 x4 x3 ⟶ not (x2 x5 x3) ⟶ x2 x6 x3 ⟶ x2 x7 x3 ⟶ not (x2 x8 x3) ⟶ not (x2 x9 x3) ⟶ not (x2 x10 x3) ⟶ x11) ⟶ (x2 x4 x3 ⟶ x2 x5 x3 ⟶ x2 x6 x3 ⟶ x2 x7 x3 ⟶ not (x2 x8 x3) ⟶ not (x2 x9 x3) ⟶ not (x2 x10 x3) ⟶ x11) ⟶ (x2 x4 x3 ⟶ not (x2 x5 x3) ⟶ not (x2 x6 x3) ⟶ not (x2 x7 x3) ⟶ x2 x8 x3 ⟶ not (x2 x9 x3) ⟶ not (x2 x10 x3) ⟶ x11) ⟶ (x2 x4 x3 ⟶ not (x2 x5 x3) ⟶ not (x2 x6 x3) ⟶ not (x2 x7 x3) ⟶ not (x2 x8 x3) ⟶ x2 x9 x3 ⟶ not (x2 x10 x3) ⟶ x11) ⟶ (x2 x4 x3 ⟶ not (x2 x5 x3) ⟶ not (x2 x6 x3) ⟶ not (x2 x7 x3) ⟶ x2 x8 x3 ⟶ x2 x9 x3 ⟶ not (x2 x10 x3) ⟶ x11) ⟶ (x2 x4 x3 ⟶ not (x2 x5 x3) ⟶ not (x2 x6 x3) ⟶ not (x2 x7 x3) ⟶ not (x2 x8 x3) ⟶ not (x2 x9 x3) ⟶ x2 x10 x3 ⟶ x11) ⟶ (x2 x4 x3 ⟶ not (x2 x5 x3) ⟶ not (x2 x6 x3) ⟶ not (x2 x7 x3) ⟶ x2 x8 x3 ⟶ not (x2 x9 x3) ⟶ x2 x10 x3 ⟶ x11) ⟶ (x2 x4 x3 ⟶ not (x2 x5 x3) ⟶ not (x2 x6 x3) ⟶ not (x2 x7 x3) ⟶ not (x2 x8 x3) ⟶ x2 x9 x3 ⟶ x2 x10 x3 ⟶ x11) ⟶ (x2 x4 x3 ⟶ not (x2 x5 x3) ⟶ not (x2 x6 x3) ⟶ not (x2 x7 x3) ⟶ x2 x8 x3 ⟶ x2 x9 x3 ⟶ x2 x10 x3 ⟶ x11) ⟶ x11 (proof)
Definition 13b0f.. := λ x0 : ι → ι → ο . λ x1 x2 x3 x4 x5 x6 x7 . ∀ x8 : ο . (9ab39.. x0 x1 x2 x3 x4 x5 x6 ⟶ (x1 = x7 ⟶ ∀ x9 : ο . x9) ⟶ (x2 = x7 ⟶ ∀ x9 : ο . x9) ⟶ (x3 = x7 ⟶ ∀ x9 : ο . x9) ⟶ (x4 = x7 ⟶ ∀ x9 : ο . x9) ⟶ (x5 = x7 ⟶ ∀ x9 : ο . x9) ⟶ (x6 = x7 ⟶ ∀ x9 : ο . x9) ⟶ x0 x1 x7 ⟶ x0 x2 x7 ⟶ x0 x3 x7 ⟶ x0 x4 x7 ⟶ not (x0 x5 x7) ⟶ not (x0 x6 x7) ⟶ x8) ⟶ x8
Theorem 8683f.. : ∀ x0 x1 . ∀ x2 : ι → ι → ο . (∀ x3 . x3 ∈ x1 ⟶ ∀ x4 . x4 ∈ x1 ⟶ x2 x3 x4 ⟶ x2 x4 x3) ⟶ 4402e.. x1 x2 ⟶ cf2df.. x1 x2 ⟶ ∀ x3 . x3 ∈ x1 ⟶ x0 ⊆ setminus x1 (Sing x3) ⟶ ∀ x4 . x4 ∈ x0 ⟶ ∀ x5 . x5 ∈ x0 ⟶ ∀ x6 . x6 ∈ x0 ⟶ ∀ x7 . x7 ∈ x0 ⟶ ∀ x8 . x8 ∈ x0 ⟶ ∀ x9 . x9 ∈ x0 ⟶ ∀ x10 . x10 ∈ x0 ⟶ 13b0f.. x2 x4 x5 x6 x7 x8 x9 x10 ⟶ ∀ x11 : ο . (x2 x4 x3 ⟶ not (x2 x5 x3) ⟶ not (x2 x6 x3) ⟶ not (x2 x7 x3) ⟶ not (x2 x8 x3) ⟶ not (x2 x9 x3) ⟶ not (x2 x10 x3) ⟶ x11) ⟶ (not (x2 x4 x3) ⟶ x2 x5 x3 ⟶ not (x2 x6 x3) ⟶ not (x2 x7 x3) ⟶ not (x2 x8 x3) ⟶ not (x2 x9 x3) ⟶ not (x2 x10 x3) ⟶ x11) ⟶ (x2 x4 x3 ⟶ x2 x5 x3 ⟶ not (x2 x6 x3) ⟶ not (x2 x7 x3) ⟶ not (x2 x8 x3) ⟶ not (x2 x9 x3) ⟶ not (x2 x10 x3) ⟶ x11) ⟶ (not (x2 x4 x3) ⟶ not (x2 x5 x3) ⟶ x2 x6 x3 ⟶ not (x2 x7 x3) ⟶ not (x2 x8 x3) ⟶ not (x2 x9 x3) ⟶ not (x2 x10 x3) ⟶ x11) ⟶ (x2 x4 x3 ⟶ not (x2 x5 x3) ⟶ x2 x6 x3 ⟶ not (x2 x7 x3) ⟶ not (x2 x8 x3) ⟶ not (x2 x9 x3) ⟶ not (x2 x10 x3) ⟶ x11) ⟶ (not (x2 x4 x3) ⟶ x2 x5 x3 ⟶ x2 x6 x3 ⟶ not (x2 x7 x3) ⟶ not (x2 x8 x3) ⟶ not (x2 x9 x3) ⟶ not (x2 x10 x3) ⟶ x11) ⟶ (x2 x4 x3 ⟶ x2 x5 x3 ⟶ x2 x6 x3 ⟶ not (x2 x7 x3) ⟶ not (x2 x8 x3) ⟶ not (x2 x9 x3) ⟶ not (x2 x10 x3) ⟶ x11) ⟶ (not (x2 x4 x3) ⟶ not (x2 x5 x3) ⟶ not (x2 x6 x3) ⟶ x2 x7 x3 ⟶ not (x2 x8 x3) ⟶ not (x2 x9 x3) ⟶ not (x2 x10 x3) ⟶ x11) ⟶ (x2 x4 x3 ⟶ not (x2 x5 x3) ⟶ not (x2 x6 x3) ⟶ x2 x7 x3 ⟶ not (x2 x8 x3) ⟶ not (x2 x9 x3) ⟶ not (x2 x10 x3) ⟶ x11) ⟶ (not (x2 x4 x3) ⟶ x2 x5 x3 ⟶ not (x2 x6 x3) ⟶ x2 x7 x3 ⟶ not (x2 x8 x3) ⟶ not (x2 x9 x3) ⟶ not (x2 x10 x3) ⟶ x11) ⟶ (x2 x4 x3 ⟶ x2 x5 x3 ⟶ not (x2 x6 x3) ⟶ x2 x7 x3 ⟶ not (x2 x8 x3) ⟶ not (x2 x9 x3) ⟶ not (x2 x10 x3) ⟶ x11) ⟶ (not (x2 x4 x3) ⟶ not (x2 x5 x3) ⟶ x2 x6 x3 ⟶ x2 x7 x3 ⟶ not (x2 x8 x3) ⟶ not (x2 x9 x3) ⟶ not (x2 x10 x3) ⟶ x11) ⟶ (x2 x4 x3 ⟶ not (x2 x5 x3) ⟶ x2 x6 x3 ⟶ x2 x7 x3 ⟶ not (x2 x8 x3) ⟶ not (x2 x9 x3) ⟶ not (x2 x10 x3) ⟶ x11) ⟶ (not (x2 x4 x3) ⟶ x2 x5 x3 ⟶ x2 x6 x3 ⟶ x2 x7 x3 ⟶ not (x2 x8 x3) ⟶ not (x2 x9 x3) ⟶ not (x2 x10 x3) ⟶ x11) ⟶ (x2 x4 x3 ⟶ x2 x5 x3 ⟶ x2 x6 x3 ⟶ x2 x7 x3 ⟶ not (x2 x8 x3) ⟶ not (x2 x9 x3) ⟶ not (x2 x10 x3) ⟶ x11) ⟶ (x2 x4 x3 ⟶ not (x2 x5 x3) ⟶ not (x2 x6 x3) ⟶ not (x2 x7 x3) ⟶ x2 x8 x3 ⟶ not (x2 x9 x3) ⟶ not (x2 x10 x3) ⟶ x11) ⟶ (x2 x4 x3 ⟶ not (x2 x5 x3) ⟶ not (x2 x6 x3) ⟶ not (x2 x7 x3) ⟶ not (x2 x8 x3) ⟶ x2 x9 x3 ⟶ not (x2 x10 x3) ⟶ x11) ⟶ (x2 x4 x3 ⟶ not (x2 x5 x3) ⟶ not (x2 x6 x3) ⟶ not (x2 x7 x3) ⟶ x2 x8 x3 ⟶ x2 x9 x3 ⟶ not (x2 x10 x3) ⟶ x11) ⟶ x11 (proof)
Definition 1e330.. := λ x0 : ι → ι → ο . λ x1 x2 x3 x4 x5 x6 . ∀ x7 : ο . (2b028.. x0 x1 x2 x3 x4 x5 ⟶ (x1 = x6 ⟶ ∀ x8 : ο . x8) ⟶ (x2 = x6 ⟶ ∀ x8 : ο . x8) ⟶ (x3 = x6 ⟶ ∀ x8 : ο . x8) ⟶ (x4 = x6 ⟶ ∀ x8 : ο . x8) ⟶ (x5 = x6 ⟶ ∀ x8 : ο . x8) ⟶ x0 x1 x6 ⟶ x0 x2 x6 ⟶ x0 x3 x6 ⟶ x0 x4 x6 ⟶ not (x0 x5 x6) ⟶ x7) ⟶ x7
Definition c8dd3.. := λ x0 : ι → ι → ο . λ x1 x2 x3 x4 x5 x6 x7 . ∀ x8 : ο . (1e330.. x0 x1 x2 x3 x4 x5 x6 ⟶ (x1 = x7 ⟶ ∀ x9 : ο . x9) ⟶ (x2 = x7 ⟶ ∀ x9 : ο . x9) ⟶ (x3 = x7 ⟶ ∀ x9 : ο . x9) ⟶ (x4 = x7 ⟶ ∀ x9 : ο . x9) ⟶ (x5 = x7 ⟶ ∀ x9 : ο . x9) ⟶ (x6 = x7 ⟶ ∀ x9 : ο . x9) ⟶ x0 x1 x7 ⟶ x0 x2 x7 ⟶ x0 x3 x7 ⟶ x0 x4 x7 ⟶ not (x0 x5 x7) ⟶ not (x0 x6 x7) ⟶ x8) ⟶ x8
Theorem a8ed5.. : ∀ x0 x1 . ∀ x2 : ι → ι → ο . (∀ x3 . x3 ∈ x1 ⟶ ∀ x4 . x4 ∈ x1 ⟶ x2 x3 x4 ⟶ x2 x4 x3) ⟶ 4402e.. x1 x2 ⟶ cf2df.. x1 x2 ⟶ ∀ x3 . x3 ∈ x1 ⟶ x0 ⊆ setminus x1 (Sing x3) ⟶ ∀ x4 . x4 ∈ x0 ⟶ ∀ x5 . x5 ∈ x0 ⟶ ∀ x6 . x6 ∈ x0 ⟶ ∀ x7 . x7 ∈ x0 ⟶ ∀ x8 . x8 ∈ x0 ⟶ ∀ x9 . x9 ∈ x0 ⟶ ∀ x10 . x10 ∈ x0 ⟶ c8dd3.. x2 x4 x5 x6 x7 x8 x9 x10 ⟶ ∀ x11 : ο . (x2 x4 x3 ⟶ not (x2 x5 x3) ⟶ not (x2 x6 x3) ⟶ not (x2 x7 x3) ⟶ not (x2 x8 x3) ⟶ not (x2 x9 x3) ⟶ not (x2 x10 x3) ⟶ x11) ⟶ (not (x2 x4 x3) ⟶ x2 x5 x3 ⟶ not (x2 x6 x3) ⟶ not (x2 x7 x3) ⟶ not (x2 x8 x3) ⟶ not (x2 x9 x3) ⟶ not (x2 x10 x3) ⟶ x11) ⟶ (x2 x4 x3 ⟶ x2 x5 x3 ⟶ not (x2 x6 x3) ⟶ not (x2 x7 x3) ⟶ not (x2 x8 x3) ⟶ not (x2 x9 x3) ⟶ not (x2 x10 x3) ⟶ x11) ⟶ (not (x2 x4 x3) ⟶ not (x2 x5 x3) ⟶ x2 x6 x3 ⟶ not (x2 x7 x3) ⟶ not (x2 x8 x3) ⟶ not (x2 x9 x3) ⟶ not (x2 x10 x3) ⟶ x11) ⟶ (x2 x4 x3 ⟶ not (x2 x5 x3) ⟶ x2 x6 x3 ⟶ not (x2 x7 x3) ⟶ not (x2 x8 x3) ⟶ not (x2 x9 x3) ⟶ not (x2 x10 x3) ⟶ x11) ⟶ (not (x2 x4 x3) ⟶ x2 x5 x3 ⟶ x2 x6 x3 ⟶ not (x2 x7 x3) ⟶ not (x2 x8 x3) ⟶ not (x2 x9 x3) ⟶ not (x2 x10 x3) ⟶ x11) ⟶ (x2 x4 x3 ⟶ x2 x5 x3 ⟶ x2 x6 x3 ⟶ not (x2 x7 x3) ⟶ not (x2 x8 x3) ⟶ not (x2 x9 x3) ⟶ not (x2 x10 x3) ⟶ x11) ⟶ (not (x2 x4 x3) ⟶ not (x2 x5 x3) ⟶ not (x2 x6 x3) ⟶ x2 x7 x3 ⟶ not (x2 x8 x3) ⟶ not (x2 x9 x3) ⟶ not (x2 x10 x3) ⟶ x11) ⟶ (x2 x4 x3 ⟶ not (x2 x5 x3) ⟶ not (x2 x6 x3) ⟶ x2 x7 x3 ⟶ not (x2 x8 x3) ⟶ not (x2 x9 x3) ⟶ not (x2 x10 x3) ⟶ x11) ⟶ (not (x2 x4 x3) ⟶ x2 x5 x3 ⟶ not (x2 x6 x3) ⟶ x2 x7 x3 ⟶ not (x2 x8 x3) ⟶ not (x2 x9 x3) ⟶ not (x2 x10 x3) ⟶ x11) ⟶ (x2 x4 x3 ⟶ x2 x5 x3 ⟶ not (x2 x6 x3) ⟶ x2 x7 x3 ⟶ not (x2 x8 x3) ⟶ not (x2 x9 x3) ⟶ not (x2 x10 x3) ⟶ x11) ⟶ (not (x2 x4 x3) ⟶ not (x2 x5 x3) ⟶ x2 x6 x3 ⟶ x2 x7 x3 ⟶ not (x2 x8 x3) ⟶ not (x2 x9 x3) ⟶ not (x2 x10 x3) ⟶ x11) ⟶ (x2 x4 x3 ⟶ not (x2 x5 x3) ⟶ x2 x6 x3 ⟶ x2 x7 x3 ⟶ not (x2 x8 x3) ⟶ not (x2 x9 x3) ⟶ not (x2 x10 x3) ⟶ x11) ⟶ (not (x2 x4 x3) ⟶ x2 x5 x3 ⟶ x2 x6 x3 ⟶ x2 x7 x3 ⟶ not (x2 x8 x3) ⟶ not (x2 x9 x3) ⟶ not (x2 x10 x3) ⟶ x11) ⟶ (x2 x4 x3 ⟶ x2 x5 x3 ⟶ x2 x6 x3 ⟶ x2 x7 x3 ⟶ not (x2 x8 x3) ⟶ not (x2 x9 x3) ⟶ not (x2 x10 x3) ⟶ x11) ⟶ (x2 x4 x3 ⟶ not (x2 x5 x3) ⟶ not (x2 x6 x3) ⟶ not (x2 x7 x3) ⟶ x2 x8 x3 ⟶ not (x2 x9 x3) ⟶ not (x2 x10 x3) ⟶ x11) ⟶ x11 (proof)
Definition 80df3.. := λ x0 : ι → ι → ο . λ x1 x2 x3 x4 x5 . ∀ x6 : ο . (8b6ad.. x0 x1 x2 x3 x4 ⟶ (x1 = x5 ⟶ ∀ x7 : ο . x7) ⟶ (x2 = x5 ⟶ ∀ x7 : ο . x7) ⟶ (x3 = x5 ⟶ ∀ x7 : ο . x7) ⟶ (x4 = x5 ⟶ ∀ x7 : ο . x7) ⟶ x0 x1 x5 ⟶ x0 x2 x5 ⟶ x0 x3 x5 ⟶ x0 x4 x5 ⟶ x6) ⟶ x6
Definition af3c4.. := λ x0 : ι → ι → ο . λ x1 x2 x3 x4 x5 x6 . ∀ x7 : ο . (80df3.. x0 x1 x2 x3 x4 x5 ⟶ (x1 = x6 ⟶ ∀ x8 : ο . x8) ⟶ (x2 = x6 ⟶ ∀ x8 : ο . x8) ⟶ (x3 = x6 ⟶ ∀ x8 : ο . x8) ⟶ (x4 = x6 ⟶ ∀ x8 : ο . x8) ⟶ (x5 = x6 ⟶ ∀ x8 : ο . x8) ⟶ x0 x1 x6 ⟶ x0 x2 x6 ⟶ x0 x3 x6 ⟶ x0 x4 x6 ⟶ not (x0 x5 x6) ⟶ x7) ⟶ x7
Definition 2bd79.. := λ x0 : ι → ι → ο . λ x1 x2 x3 x4 x5 x6 x7 . ∀ x8 : ο . (af3c4.. x0 x1 x2 x3 x4 x5 x6 ⟶ (x1 = x7 ⟶ ∀ x9 : ο . x9) ⟶ (x2 = x7 ⟶ ∀ x9 : ο . x9) ⟶ (x3 = x7 ⟶ ∀ x9 : ο . x9) ⟶ (x4 = x7 ⟶ ∀ x9 : ο . x9) ⟶ (x5 = x7 ⟶ ∀ x9 : ο . x9) ⟶ (x6 = x7 ⟶ ∀ x9 : ο . x9) ⟶ x0 x1 x7 ⟶ x0 x2 x7 ⟶ x0 x3 x7 ⟶ x0 x4 x7 ⟶ not (x0 x5 x7) ⟶ not (x0 x6 x7) ⟶ x8) ⟶ x8
Theorem 0230d.. : ∀ x0 x1 . ∀ x2 : ι → ι → ο . (∀ x3 . x3 ∈ x1 ⟶ ∀ x4 . x4 ∈ x1 ⟶ x2 x3 x4 ⟶ x2 x4 x3) ⟶ 4402e.. x1 x2 ⟶ cf2df.. x1 x2 ⟶ ∀ x3 . x3 ∈ x1 ⟶ x0 ⊆ setminus x1 (Sing x3) ⟶ ∀ x4 . x4 ∈ x0 ⟶ ∀ x5 . x5 ∈ x0 ⟶ ∀ x6 . x6 ∈ x0 ⟶ ∀ x7 . x7 ∈ x0 ⟶ ∀ x8 . x8 ∈ x0 ⟶ ∀ x9 . x9 ∈ x0 ⟶ ∀ x10 . x10 ∈ x0 ⟶ 2bd79.. x2 x4 x5 x6 x7 x8 x9 x10 ⟶ ∀ x11 : ο . (x2 x4 x3 ⟶ not (x2 x5 x3) ⟶ not (x2 x6 x3) ⟶ not (x2 x7 x3) ⟶ not (x2 x8 x3) ⟶ not (x2 x9 x3) ⟶ not (x2 x10 x3) ⟶ x11) ⟶ (not (x2 x4 x3) ⟶ x2 x5 x3 ⟶ not (x2 x6 x3) ⟶ not (x2 x7 x3) ⟶ not (x2 x8 x3) ⟶ not (x2 x9 x3) ⟶ not (x2 x10 x3) ⟶ x11) ⟶ (x2 x4 x3 ⟶ x2 x5 x3 ⟶ not (x2 x6 x3) ⟶ not (x2 x7 x3) ⟶ not (x2 x8 x3) ⟶ not (x2 x9 x3) ⟶ not (x2 x10 x3) ⟶ x11) ⟶ (not (x2 x4 x3) ⟶ not (x2 x5 x3) ⟶ x2 x6 x3 ⟶ not (x2 x7 x3) ⟶ not (x2 x8 x3) ⟶ not (x2 x9 x3) ⟶ not (x2 x10 x3) ⟶ x11) ⟶ (x2 x4 x3 ⟶ not (x2 x5 x3) ⟶ x2 x6 x3 ⟶ not (x2 x7 x3) ⟶ not (x2 x8 x3) ⟶ not (x2 x9 x3) ⟶ not (x2 x10 x3) ⟶ x11) ⟶ (not (x2 x4 x3) ⟶ x2 x5 x3 ⟶ x2 x6 x3 ⟶ not (x2 x7 x3) ⟶ not (x2 x8 x3) ⟶ not (x2 x9 x3) ⟶ not (x2 x10 x3) ⟶ x11) ⟶ (x2 x4 x3 ⟶ x2 x5 x3 ⟶ x2 x6 x3 ⟶ not (x2 x7 x3) ⟶ not (x2 x8 x3) ⟶ not (x2 x9 x3) ⟶ not (x2 x10 x3) ⟶ x11) ⟶ (not (x2 x4 x3) ⟶ not (x2 x5 x3) ⟶ not (x2 x6 x3) ⟶ x2 x7 x3 ⟶ not (x2 x8 x3) ⟶ not (x2 x9 x3) ⟶ not (x2 x10 x3) ⟶ x11) ⟶ (x2 x4 x3 ⟶ not (x2 x5 x3) ⟶ not (x2 x6 x3) ⟶ x2 x7 x3 ⟶ not (x2 x8 x3) ⟶ not (x2 x9 x3) ⟶ not (x2 x10 x3) ⟶ x11) ⟶ (not (x2 x4 x3) ⟶ x2 x5 x3 ⟶ not (x2 x6 x3) ⟶ x2 x7 x3 ⟶ not (x2 x8 x3) ⟶ not (x2 x9 x3) ⟶ not (x2 x10 x3) ⟶ x11) ⟶ (x2 x4 x3 ⟶ x2 x5 x3 ⟶ not (x2 x6 x3) ⟶ x2 x7 x3 ⟶ not (x2 x8 x3) ⟶ not (x2 x9 x3) ⟶ not (x2 x10 x3) ⟶ x11) ⟶ (not (x2 x4 x3) ⟶ not (x2 x5 x3) ⟶ x2 x6 x3 ⟶ x2 x7 x3 ⟶ not (x2 x8 x3) ⟶ not (x2 x9 x3) ⟶ not (x2 x10 x3) ⟶ x11) ⟶ (x2 x4 x3 ⟶ not (x2 x5 x3) ⟶ x2 x6 x3 ⟶ x2 x7 x3 ⟶ not (x2 x8 x3) ⟶ not (x2 x9 x3) ⟶ not (x2 x10 x3) ⟶ x11) ⟶ (not (x2 x4 x3) ⟶ x2 x5 x3 ⟶ x2 x6 x3 ⟶ x2 x7 x3 ⟶ not (x2 x8 x3) ⟶ not (x2 x9 x3) ⟶ not (x2 x10 x3) ⟶ x11) ⟶ (x2 x4 x3 ⟶ x2 x5 x3 ⟶ x2 x6 x3 ⟶ x2 x7 x3 ⟶ not (x2 x8 x3) ⟶ not (x2 x9 x3) ⟶ not (x2 x10 x3) ⟶ x11) ⟶ x11 (proof)
Definition 62523.. := λ x0 : ι → ι → ο . λ x1 x2 x3 x4 x5 . ∀ x6 : ο . (8b6ad.. x0 x1 x2 x3 x4 ⟶ (x1 = x5 ⟶ ∀ x7 : ο . x7) ⟶ (x2 = x5 ⟶ ∀ x7 : ο . x7) ⟶ (x3 = x5 ⟶ ∀ x7 : ο . x7) ⟶ (x4 = x5 ⟶ ∀ x7 : ο . x7) ⟶ not (x0 x1 x5) ⟶ not (x0 x2 x5) ⟶ not (x0 x3 x5) ⟶ x0 x4 x5 ⟶ x6) ⟶ x6
Definition a542b.. := λ x0 : ι → ι → ο . λ x1 x2 x3 x4 x5 x6 . ∀ x7 : ο . (62523.. x0 x1 x2 x3 x4 x5 ⟶ (x1 = x6 ⟶ ∀ x8 : ο . x8) ⟶ (x2 = x6 ⟶ ∀ x8 : ο . x8) ⟶ (x3 = x6 ⟶ ∀ x8 : ο . x8) ⟶ (x4 = x6 ⟶ ∀ x8 : ο . x8) ⟶ (x5 = x6 ⟶ ∀ x8 : ο . x8) ⟶ not (x0 x1 x6) ⟶ not (x0 x2 x6) ⟶ x0 x3 x6 ⟶ not (x0 x4 x6) ⟶ x0 x5 x6 ⟶ x7) ⟶ x7
Definition 2fb86.. := λ x0 : ι → ι → ο . λ x1 x2 x3 x4 x5 x6 x7 . ∀ x8 : ο . (a542b.. x0 x1 x2 x3 x4 x5 x6 ⟶ (x1 = x7 ⟶ ∀ x9 : ο . x9) ⟶ (x2 = x7 ⟶ ∀ x9 : ο . x9) ⟶ (x3 = x7 ⟶ ∀ x9 : ο . x9) ⟶ (x4 = x7 ⟶ ∀ x9 : ο . x9) ⟶ (x5 = x7 ⟶ ∀ x9 : ο . x9) ⟶ (x6 = x7 ⟶ ∀ x9 : ο . x9) ⟶ x0 x1 x7 ⟶ x0 x2 x7 ⟶ not (x0 x3 x7) ⟶ not (x0 x4 x7) ⟶ not (x0 x5 x7) ⟶ not (x0 x6 x7) ⟶ x8) ⟶ x8
Theorem 34d1d.. : ∀ x0 x1 . ∀ x2 : ι → ι → ο . (∀ x3 . x3 ∈ x1 ⟶ ∀ x4 . x4 ∈ x1 ⟶ x2 x3 x4 ⟶ x2 x4 x3) ⟶ 4402e.. x1 x2 ⟶ cf2df.. x1 x2 ⟶ ∀ x3 . x3 ∈ x1 ⟶ x0 ⊆ setminus x1 (Sing x3) ⟶ ∀ x4 . x4 ∈ x0 ⟶ ∀ x5 . x5 ∈ x0 ⟶ ∀ x6 . x6 ∈ x0 ⟶ ∀ x7 . x7 ∈ x0 ⟶ ∀ x8 . x8 ∈ x0 ⟶ ∀ x9 . x9 ∈ x0 ⟶ ∀ x10 . x10 ∈ x0 ⟶ 2fb86.. x2 x4 x5 x6 x7 x8 x9 x10 ⟶ ∀ x11 : ο . (x2 x4 x3 ⟶ not (x2 x5 x3) ⟶ not (x2 x6 x3) ⟶ not (x2 x7 x3) ⟶ not (x2 x8 x3) ⟶ not (x2 x9 x3) ⟶ not (x2 x10 x3) ⟶ x11) ⟶ (not (x2 x4 x3) ⟶ x2 x5 x3 ⟶ not (x2 x6 x3) ⟶ not (x2 x7 x3) ⟶ not (x2 x8 x3) ⟶ not (x2 x9 x3) ⟶ not (x2 x10 x3) ⟶ x11) ⟶ (x2 x4 x3 ⟶ x2 x5 x3 ⟶ not (x2 x6 x3) ⟶ not (x2 x7 x3) ⟶ not (x2 x8 x3) ⟶ not (x2 x9 x3) ⟶ not (x2 x10 x3) ⟶ x11) ⟶ (x2 x4 x3 ⟶ not (x2 x5 x3) ⟶ x2 x6 x3 ⟶ not (x2 x7 x3) ⟶ not (x2 x8 x3) ⟶ not (x2 x9 x3) ⟶ not (x2 x10 x3) ⟶ x11) ⟶ (not (x2 x4 x3) ⟶ x2 x5 x3 ⟶ x2 x6 x3 ⟶ not (x2 x7 x3) ⟶ not (x2 x8 x3) ⟶ not (x2 x9 x3) ⟶ not (x2 x10 x3) ⟶ x11) ⟶ (x2 x4 x3 ⟶ x2 x5 x3 ⟶ x2 x6 x3 ⟶ not (x2 x7 x3) ⟶ not (x2 x8 x3) ⟶ not (x2 x9 x3) ⟶ not (x2 x10 x3) ⟶ x11) ⟶ (x2 x4 x3 ⟶ not (x2 x5 x3) ⟶ not (x2 x6 x3) ⟶ x2 x7 x3 ⟶ not (x2 x8 x3) ⟶ not (x2 x9 x3) ⟶ not (x2 x10 x3) ⟶ x11) ⟶ (not (x2 x4 x3) ⟶ x2 x5 x3 ⟶ not (x2 x6 x3) ⟶ x2 x7 x3 ⟶ not (x2 x8 x3) ⟶ not (x2 x9 x3) ⟶ not (x2 x10 x3) ⟶ x11) ⟶ (x2 x4 x3 ⟶ x2 x5 x3 ⟶ not (x2 x6 x3) ⟶ x2 x7 x3 ⟶ not (x2 x8 x3) ⟶ not (x2 x9 x3) ⟶ not (x2 x10 x3) ⟶ x11) ⟶ (not (x2 x4 x3) ⟶ not (x2 x5 x3) ⟶ x2 x6 x3 ⟶ x2 x7 x3 ⟶ not (x2 x8 x3) ⟶ not (x2 x9 x3) ⟶ not (x2 x10 x3) ⟶ x11) ⟶ (x2 x4 x3 ⟶ not (x2 x5 x3) ⟶ x2 x6 x3 ⟶ x2 x7 x3 ⟶ not (x2 x8 x3) ⟶ not (x2 x9 x3) ⟶ not (x2 x10 x3) ⟶ x11) ⟶ (not (x2 x4 x3) ⟶ x2 x5 x3 ⟶ x2 x6 x3 ⟶ x2 x7 x3 ⟶ not (x2 x8 x3) ⟶ not (x2 x9 x3) ⟶ not (x2 x10 x3) ⟶ x11) ⟶ (x2 x4 x3 ⟶ x2 x5 x3 ⟶ x2 x6 x3 ⟶ x2 x7 x3 ⟶ not (x2 x8 x3) ⟶ not (x2 x9 x3) ⟶ not (x2 x10 x3) ⟶ x11) ⟶ (x2 x4 x3 ⟶ not (x2 x5 x3) ⟶ not (x2 x6 x3) ⟶ not (x2 x7 x3) ⟶ x2 x8 x3 ⟶ not (x2 x9 x3) ⟶ not (x2 x10 x3) ⟶ x11) ⟶ (not (x2 x4 x3) ⟶ x2 x5 x3 ⟶ not (x2 x6 x3) ⟶ not (x2 x7 x3) ⟶ x2 x8 x3 ⟶ not (x2 x9 x3) ⟶ not (x2 x10 x3) ⟶ x11) ⟶ (x2 x4 x3 ⟶ x2 x5 x3 ⟶ not (x2 x6 x3) ⟶ not (x2 x7 x3) ⟶ x2 x8 x3 ⟶ not (x2 x9 x3) ⟶ not (x2 x10 x3) ⟶ x11) ⟶ (x2 x4 x3 ⟶ not (x2 x5 x3) ⟶ x2 x6 x3 ⟶ not (x2 x7 x3) ⟶ x2 x8 x3 ⟶ not (x2 x9 x3) ⟶ not (x2 x10 x3) ⟶ x11) ⟶ (not (x2 x4 x3) ⟶ x2 x5 x3 ⟶ x2 x6 x3 ⟶ not (x2 x7 x3) ⟶ x2 x8 x3 ⟶ not (x2 x9 x3) ⟶ not (x2 x10 x3) ⟶ x11) ⟶ (x2 x4 x3 ⟶ x2 x5 x3 ⟶ x2 x6 x3 ⟶ not (x2 x7 x3) ⟶ x2 x8 x3 ⟶ not (x2 x9 x3) ⟶ not (x2 x10 x3) ⟶ x11) ⟶ (x2 x4 x3 ⟶ not (x2 x5 x3) ⟶ not (x2 x6 x3) ⟶ not (x2 x7 x3) ⟶ not (x2 x8 x3) ⟶ x2 x9 x3 ⟶ not (x2 x10 x3) ⟶ x11) ⟶ (not (x2 x4 x3) ⟶ x2 x5 x3 ⟶ not (x2 x6 x3) ⟶ not (x2 x7 x3) ⟶ not (x2 x8 x3) ⟶ x2 x9 x3 ⟶ not (x2 x10 x3) ⟶ x11) ⟶ (x2 x4 x3 ⟶ x2 x5 x3 ⟶ not (x2 x6 x3) ⟶ not (x2 x7 x3) ⟶ not (x2 x8 x3) ⟶ x2 x9 x3 ⟶ not (x2 x10 x3) ⟶ x11) ⟶ (x2 x4 x3 ⟶ not (x2 x5 x3) ⟶ not (x2 x6 x3) ⟶ x2 x7 x3 ⟶ not (x2 x8 x3) ⟶ x2 x9 x3 ⟶ not (x2 x10 x3) ⟶ x11) ⟶ (not (x2 x4 x3) ⟶ x2 x5 x3 ⟶ not (x2 x6 x3) ⟶ x2 x7 x3 ⟶ not (x2 x8 x3) ⟶ x2 x9 x3 ⟶ not (x2 x10 x3) ⟶ x11) ⟶ (x2 x4 x3 ⟶ x2 x5 x3 ⟶ not (x2 x6 x3) ⟶ x2 x7 x3 ⟶ not (x2 x8 x3) ⟶ x2 x9 x3 ⟶ not (x2 x10 x3) ⟶ x11) ⟶ (not (x2 x4 x3) ⟶ not (x2 x5 x3) ⟶ x2 x6 x3 ⟶ x2 x7 x3 ⟶ not (x2 x8 x3) ⟶ not (x2 x9 x3) ⟶ x2 x10 x3 ⟶ x11) ⟶ x11 (proof)
Definition 87c36.. := λ x0 : ι → ι → ο . λ x1 x2 x3 x4 x5 . ∀ x6 : ο . (2f869.. x0 x1 x2 x3 x4 ⟶ (x1 = x5 ⟶ ∀ x7 : ο . x7) ⟶ (x2 = x5 ⟶ ∀ x7 : ο . x7) ⟶ (x3 = x5 ⟶ ∀ x7 : ο . x7) ⟶ (x4 = x5 ⟶ ∀ x7 : ο . x7) ⟶ not (x0 x1 x5) ⟶ x0 x2 x5 ⟶ not (x0 x3 x5) ⟶ x0 x4 x5 ⟶ x6) ⟶ x6
Definition f201d.. := λ x0 : ι → ι → ο . λ x1 x2 x3 x4 x5 x6 . ∀ x7 : ο . (87c36.. x0 x1 x2 x3 x4 x5 ⟶ (x1 = x6 ⟶ ∀ x8 : ο . x8) ⟶ (x2 = x6 ⟶ ∀ x8 : ο . x8) ⟶ (x3 = x6 ⟶ ∀ x8 : ο . x8) ⟶ (x4 = x6 ⟶ ∀ x8 : ο . x8) ⟶ (x5 = x6 ⟶ ∀ x8 : ο . x8) ⟶ x0 x1 x6 ⟶ not (x0 x2 x6) ⟶ x0 x3 x6 ⟶ not (x0 x4 x6) ⟶ x0 x5 x6 ⟶ x7) ⟶ x7
Definition 81638.. := λ x0 : ι → ι → ο . λ x1 x2 x3 x4 x5 x6 x7 . ∀ x8 : ο . (f201d.. x0 x1 x2 x3 x4 x5 x6 ⟶ (x1 = x7 ⟶ ∀ x9 : ο . x9) ⟶ (x2 = x7 ⟶ ∀ x9 : ο . x9) ⟶ (x3 = x7 ⟶ ∀ x9 : ο . x9) ⟶ (x4 = x7 ⟶ ∀ x9 : ο . x9) ⟶ (x5 = x7 ⟶ ∀ x9 : ο . x9) ⟶ (x6 = x7 ⟶ ∀ x9 : ο . x9) ⟶ x0 x1 x7 ⟶ x0 x2 x7 ⟶ not (x0 x3 x7) ⟶ not (x0 x4 x7) ⟶ not (x0 x5 x7) ⟶ not (x0 x6 x7) ⟶ x8) ⟶ x8
Theorem b246d.. : ∀ x0 x1 . ∀ x2 : ι → ι → ο . (∀ x3 . x3 ∈ x1 ⟶ ∀ x4 . x4 ∈ x1 ⟶ x2 x3 x4 ⟶ x2 x4 x3) ⟶ 4402e.. x1 x2 ⟶ cf2df.. x1 x2 ⟶ ∀ x3 . x3 ∈ x1 ⟶ x0 ⊆ setminus x1 (Sing x3) ⟶ ∀ x4 . x4 ∈ x0 ⟶ ∀ x5 . x5 ∈ x0 ⟶ ∀ x6 . x6 ∈ x0 ⟶ ∀ x7 . x7 ∈ x0 ⟶ ∀ x8 . x8 ∈ x0 ⟶ ∀ x9 . x9 ∈ x0 ⟶ ∀ x10 . x10 ∈ x0 ⟶ 81638.. x2 x4 x5 x6 x7 x8 x9 x10 ⟶ ∀ x11 : ο . (not (x2 x4 x3) ⟶ not (x2 x5 x3) ⟶ not (x2 x6 x3) ⟶ not (x2 x7 x3) ⟶ not (x2 x8 x3) ⟶ not (x2 x9 x3) ⟶ not (x2 x10 x3) ⟶ x11) ⟶ (x2 x4 x3 ⟶ not (x2 x5 x3) ⟶ not (x2 x6 x3) ⟶ not (x2 x7 x3) ⟶ not (x2 x8 x3) ⟶ not (x2 x9 x3) ⟶ not (x2 x10 x3) ⟶ x11) ⟶ (not (x2 x4 x3) ⟶ x2 x5 x3 ⟶ not (x2 x6 x3) ⟶ not (x2 x7 x3) ⟶ not (x2 x8 x3) ⟶ not (x2 x9 x3) ⟶ not (x2 x10 x3) ⟶ x11) ⟶ (x2 x4 x3 ⟶ x2 x5 x3 ⟶ not (x2 x6 x3) ⟶ not (x2 x7 x3) ⟶ not (x2 x8 x3) ⟶ not (x2 x9 x3) ⟶ not (x2 x10 x3) ⟶ x11) ⟶ (not (x2 x4 x3) ⟶ not (x2 x5 x3) ⟶ x2 x6 x3 ⟶ not (x2 x7 x3) ⟶ not (x2 x8 x3) ⟶ not (x2 x9 x3) ⟶ not (x2 x10 x3) ⟶ x11) ⟶ (x2 x4 x3 ⟶ not (x2 x5 x3) ⟶ x2 x6 x3 ⟶ not (x2 x7 x3) ⟶ not (x2 x8 x3) ⟶ not (x2 x9 x3) ⟶ not (x2 x10 x3) ⟶ x11) ⟶ (not (x2 x4 x3) ⟶ x2 x5 x3 ⟶ x2 x6 x3 ⟶ not (x2 x7 x3) ⟶ not (x2 x8 x3) ⟶ not (x2 x9 x3) ⟶ not (x2 x10 x3) ⟶ x11) ⟶ (x2 x4 x3 ⟶ x2 x5 x3 ⟶ x2 x6 x3 ⟶ not (x2 x7 x3) ⟶ not (x2 x8 x3) ⟶ not (x2 x9 x3) ⟶ not (x2 x10 x3) ⟶ x11) ⟶ (not (x2 x4 x3) ⟶ not (x2 x5 x3) ⟶ not (x2 x6 x3) ⟶ x2 x7 x3 ⟶ not (x2 x8 x3) ⟶ not (x2 x9 x3) ⟶ not (x2 x10 x3) ⟶ x11) ⟶ (x2 x4 x3 ⟶ not (x2 x5 x3) ⟶ not (x2 x6 x3) ⟶ x2 x7 x3 ⟶ not (x2 x8 x3) ⟶ not (x2 x9 x3) ⟶ not (x2 x10 x3) ⟶ x11) ⟶ (not (x2 x4 x3) ⟶ x2 x5 x3 ⟶ not (x2 x6 x3) ⟶ x2 x7 x3 ⟶ not (x2 x8 x3) ⟶ not (x2 x9 x3) ⟶ not (x2 x10 x3) ⟶ x11) ⟶ (x2 x4 x3 ⟶ x2 x5 x3 ⟶ not (x2 x6 x3) ⟶ x2 x7 x3 ⟶ not (x2 x8 x3) ⟶ not (x2 x9 x3) ⟶ not (x2 x10 x3) ⟶ x11) ⟶ (not (x2 x4 x3) ⟶ not (x2 x5 x3) ⟶ not (x2 x6 x3) ⟶ not (x2 x7 x3) ⟶ x2 x8 x3 ⟶ not (x2 x9 x3) ⟶ not (x2 x10 x3) ⟶ x11) ⟶ (x2 x4 x3 ⟶ not (x2 x5 x3) ⟶ not (x2 x6 x3) ⟶ not (x2 x7 x3) ⟶ x2 x8 x3 ⟶ not (x2 x9 x3) ⟶ not (x2 x10 x3) ⟶ x11) ⟶ (not (x2 x4 x3) ⟶ not (x2 x5 x3) ⟶ x2 x6 x3 ⟶ not (x2 x7 x3) ⟶ x2 x8 x3 ⟶ not (x2 x9 x3) ⟶ not (x2 x10 x3) ⟶ x11) ⟶ (x2 x4 x3 ⟶ not (x2 x5 x3) ⟶ x2 x6 x3 ⟶ not (x2 x7 x3) ⟶ x2 x8 x3 ⟶ not (x2 x9 x3) ⟶ not (x2 x10 x3) ⟶ x11) ⟶ (not (x2 x4 x3) ⟶ not (x2 x5 x3) ⟶ not (x2 x6 x3) ⟶ not (x2 x7 x3) ⟶ not (x2 x8 x3) ⟶ x2 x9 x3 ⟶ not (x2 x10 x3) ⟶ x11) ⟶ (not (x2 x4 x3) ⟶ x2 x5 x3 ⟶ not (x2 x6 x3) ⟶ not (x2 x7 x3) ⟶ not (x2 x8 x3) ⟶ x2 x9 x3 ⟶ not (x2 x10 x3) ⟶ x11) ⟶ (not (x2 x4 x3) ⟶ not (x2 x5 x3) ⟶ not (x2 x6 x3) ⟶ x2 x7 x3 ⟶ not (x2 x8 x3) ⟶ x2 x9 x3 ⟶ not (x2 x10 x3) ⟶ x11) ⟶ (not (x2 x4 x3) ⟶ x2 x5 x3 ⟶ not (x2 x6 x3) ⟶ x2 x7 x3 ⟶ not (x2 x8 x3) ⟶ x2 x9 x3 ⟶ not (x2 x10 x3) ⟶ x11) ⟶ (not (x2 x4 x3) ⟶ not (x2 x5 x3) ⟶ not (x2 x6 x3) ⟶ not (x2 x7 x3) ⟶ not (x2 x8 x3) ⟶ not (x2 x9 x3) ⟶ x2 x10 x3 ⟶ x11) ⟶ (not (x2 x4 x3) ⟶ not (x2 x5 x3) ⟶ x2 x6 x3 ⟶ not (x2 x7 x3) ⟶ not (x2 x8 x3) ⟶ not (x2 x9 x3) ⟶ x2 x10 x3 ⟶ x11) ⟶ (not (x2 x4 x3) ⟶ not (x2 x5 x3) ⟶ not (x2 x6 x3) ⟶ x2 x7 x3 ⟶ not (x2 x8 x3) ⟶ not (x2 x9 x3) ⟶ x2 x10 x3 ⟶ x11) ⟶ (not (x2 x4 x3) ⟶ not (x2 x5 x3) ⟶ not (x2 x6 x3) ⟶ not (x2 x7 x3) ⟶ x2 x8 x3 ⟶ not (x2 x9 x3) ⟶ x2 x10 x3 ⟶ x11) ⟶ (not (x2 x4 x3) ⟶ not (x2 x5 x3) ⟶ x2 x6 x3 ⟶ not (x2 x7 x3) ⟶ x2 x8 x3 ⟶ not (x2 x9 x3) ⟶ x2 x10 x3 ⟶ x11) ⟶ (not (x2 x4 x3) ⟶ not (x2 x5 x3) ⟶ not (x2 x6 x3) ⟶ not (x2 x7 x3) ⟶ not (x2 x8 x3) ⟶ x2 x9 x3 ⟶ x2 x10 x3 ⟶ x11) ⟶ (not (x2 x4 x3) ⟶ not (x2 x5 x3) ⟶ not (x2 x6 x3) ⟶ x2 x7 x3 ⟶ not (x2 x8 x3) ⟶ x2 x9 x3 ⟶ x2 x10 x3 ⟶ x11) ⟶ x11 (proof)
Theorem bb19a.. : ∀ x0 x1 . ∀ x2 : ι → ι → ο . (∀ x3 . x3 ∈ x1 ⟶ ∀ x4 . x4 ∈ x1 ⟶ x2 x3 x4 ⟶ x2 x4 x3) ⟶ 4402e.. x1 x2 ⟶ cf2df.. x1 x2 ⟶ ∀ x3 . x3 ∈ x1 ⟶ x0 ⊆ setminus x1 (Sing x3) ⟶ ∀ x4 . x4 ∈ x0 ⟶ ∀ x5 . x5 ∈ x0 ⟶ ∀ x6 . x6 ∈ x0 ⟶ ∀ x7 . x7 ∈ x0 ⟶ ∀ x8 . x8 ∈ x0 ⟶ ∀ x9 . x9 ∈ x0 ⟶ ∀ x10 . x10 ∈ x0 ⟶ 70d65.. x2 x4 x5 x6 x7 x8 x9 x10 ⟶ ∀ x11 : ο . (not (x2 x4 x3) ⟶ not (x2 x5 x3) ⟶ not (x2 x6 x3) ⟶ not (x2 x7 x3) ⟶ not (x2 x8 x3) ⟶ not (x2 x9 x3) ⟶ not (x2 x10 x3) ⟶ x11) ⟶ (x2 x4 x3 ⟶ not (x2 x5 x3) ⟶ not (x2 x6 x3) ⟶ not (x2 x7 x3) ⟶ not (x2 x8 x3) ⟶ not (x2 x9 x3) ⟶ not (x2 x10 x3) ⟶ x11) ⟶ (not (x2 x4 x3) ⟶ x2 x5 x3 ⟶ not (x2 x6 x3) ⟶ not (x2 x7 x3) ⟶ not (x2 x8 x3) ⟶ not (x2 x9 x3) ⟶ not (x2 x10 x3) ⟶ x11) ⟶ (x2 x4 x3 ⟶ x2 x5 x3 ⟶ not (x2 x6 x3) ⟶ not (x2 x7 x3) ⟶ not (x2 x8 x3) ⟶ not (x2 x9 x3) ⟶ not (x2 x10 x3) ⟶ x11) ⟶ (not (x2 x4 x3) ⟶ not (x2 x5 x3) ⟶ x2 x6 x3 ⟶ not (x2 x7 x3) ⟶ not (x2 x8 x3) ⟶ not (x2 x9 x3) ⟶ not (x2 x10 x3) ⟶ x11) ⟶ (x2 x4 x3 ⟶ not (x2 x5 x3) ⟶ x2 x6 x3 ⟶ not (x2 x7 x3) ⟶ not (x2 x8 x3) ⟶ not (x2 x9 x3) ⟶ not (x2 x10 x3) ⟶ x11) ⟶ (not (x2 x4 x3) ⟶ x2 x5 x3 ⟶ x2 x6 x3 ⟶ not (x2 x7 x3) ⟶ not (x2 x8 x3) ⟶ not (x2 x9 x3) ⟶ not (x2 x10 x3) ⟶ x11) ⟶ (x2 x4 x3 ⟶ x2 x5 x3 ⟶ x2 x6 x3 ⟶ not (x2 x7 x3) ⟶ not (x2 x8 x3) ⟶ not (x2 x9 x3) ⟶ not (x2 x10 x3) ⟶ x11) ⟶ (not (x2 x4 x3) ⟶ not (x2 x5 x3) ⟶ not (x2 x6 x3) ⟶ x2 x7 x3 ⟶ not (x2 x8 x3) ⟶ not (x2 x9 x3) ⟶ not (x2 x10 x3) ⟶ x11) ⟶ (x2 x4 x3 ⟶ not (x2 x5 x3) ⟶ not (x2 x6 x3) ⟶ x2 x7 x3 ⟶ not (x2 x8 x3) ⟶ not (x2 x9 x3) ⟶ not (x2 x10 x3) ⟶ x11) ⟶ (not (x2 x4 x3) ⟶ x2 x5 x3 ⟶ not (x2 x6 x3) ⟶ x2 x7 x3 ⟶ not (x2 x8 x3) ⟶ not (x2 x9 x3) ⟶ not (x2 x10 x3) ⟶ x11) ⟶ (x2 x4 x3 ⟶ x2 x5 x3 ⟶ not (x2 x6 x3) ⟶ x2 x7 x3 ⟶ not (x2 x8 x3) ⟶ not (x2 x9 x3) ⟶ not (x2 x10 x3) ⟶ x11) ⟶ (not (x2 x4 x3) ⟶ not (x2 x5 x3) ⟶ not (x2 x6 x3) ⟶ not (x2 x7 x3) ⟶ x2 x8 x3 ⟶ not (x2 x9 x3) ⟶ not (x2 x10 x3) ⟶ x11) ⟶ (x2 x4 x3 ⟶ not (x2 x5 x3) ⟶ not (x2 x6 x3) ⟶ not (x2 x7 x3) ⟶ x2 x8 x3 ⟶ not (x2 x9 x3) ⟶ not (x2 x10 x3) ⟶ x11) ⟶ (not (x2 x4 x3) ⟶ not (x2 x5 x3) ⟶ x2 x6 x3 ⟶ not (x2 x7 x3) ⟶ x2 x8 x3 ⟶ not (x2 x9 x3) ⟶ not (x2 x10 x3) ⟶ x11) ⟶ (x2 x4 x3 ⟶ not (x2 x5 x3) ⟶ x2 x6 x3 ⟶ not (x2 x7 x3) ⟶ x2 x8 x3 ⟶ not (x2 x9 x3) ⟶ not (x2 x10 x3) ⟶ x11) ⟶ (not (x2 x4 x3) ⟶ not (x2 x5 x3) ⟶ not (x2 x6 x3) ⟶ x2 x7 x3 ⟶ x2 x8 x3 ⟶ not (x2 x9 x3) ⟶ not (x2 x10 x3) ⟶ x11) ⟶ (x2 x4 x3 ⟶ not (x2 x5 x3) ⟶ not (x2 x6 x3) ⟶ x2 x7 x3 ⟶ x2 x8 x3 ⟶ not (x2 x9 x3) ⟶ not (x2 x10 x3) ⟶ x11) ⟶ (not (x2 x4 x3) ⟶ not (x2 x5 x3) ⟶ not (x2 x6 x3) ⟶ not (x2 x7 x3) ⟶ not (x2 x8 x3) ⟶ x2 x9 x3 ⟶ not (x2 x10 x3) ⟶ x11) ⟶ (not (x2 x4 x3) ⟶ x2 x5 x3 ⟶ not (x2 x6 x3) ⟶ not (x2 x7 x3) ⟶ not (x2 x8 x3) ⟶ x2 x9 x3 ⟶ not (x2 x10 x3) ⟶ x11) ⟶ (not (x2 x4 x3) ⟶ not (x2 x5 x3) ⟶ x2 x6 x3 ⟶ not (x2 x7 x3) ⟶ not (x2 x8 x3) ⟶ x2 x9 x3 ⟶ not (x2 x10 x3) ⟶ x11) ⟶ (not (x2 x4 x3) ⟶ x2 x5 x3 ⟶ x2 x6 x3 ⟶ not (x2 x7 x3) ⟶ not (x2 x8 x3) ⟶ x2 x9 x3 ⟶ not (x2 x10 x3) ⟶ x11) ⟶ (not (x2 x4 x3) ⟶ not (x2 x5 x3) ⟶ not (x2 x6 x3) ⟶ not (x2 x7 x3) ⟶ not (x2 x8 x3) ⟶ not (x2 x9 x3) ⟶ x2 x10 x3 ⟶ x11) ⟶ (not (x2 x4 x3) ⟶ not (x2 x5 x3) ⟶ not (x2 x6 x3) ⟶ x2 x7 x3 ⟶ not (x2 x8 x3) ⟶ not (x2 x9 x3) ⟶ x2 x10 x3 ⟶ x11) ⟶ (not (x2 x4 x3) ⟶ not (x2 x5 x3) ⟶ not (x2 x6 x3) ⟶ not (x2 x7 x3) ⟶ x2 x8 x3 ⟶ not (x2 x9 x3) ⟶ x2 x10 x3 ⟶ x11) ⟶ (not (x2 x4 x3) ⟶ not (x2 x5 x3) ⟶ not (x2 x6 x3) ⟶ x2 x7 x3 ⟶ x2 x8 x3 ⟶ not (x2 x9 x3) ⟶ x2 x10 x3 ⟶ x11) ⟶ (not (x2 x4 x3) ⟶ not (x2 x5 x3) ⟶ not (x2 x6 x3) ⟶ not (x2 x7 x3) ⟶ not (x2 x8 x3) ⟶ x2 x9 x3 ⟶ x2 x10 x3 ⟶ x11) ⟶ x11 (proof)
Theorem 702bd.. : ∀ x0 x1 . ∀ x2 : ι → ι → ο . (∀ x3 . x3 ∈ x1 ⟶ ∀ x4 . x4 ∈ x1 ⟶ x2 x3 x4 ⟶ x2 x4 x3) ⟶ 4402e.. x1 x2 ⟶ cf2df.. x1 x2 ⟶ ∀ x3 . x3 ∈ x1 ⟶ x0 ⊆ setminus x1 (Sing x3) ⟶ ∀ x4 . x4 ∈ x0 ⟶ ∀ x5 . x5 ∈ x0 ⟶ ∀ x6 . x6 ∈ x0 ⟶ ∀ x7 . x7 ∈ x0 ⟶ ∀ x8 . x8 ∈ x0 ⟶ ∀ x9 . x9 ∈ x0 ⟶ ∀ x10 . x10 ∈ x0 ⟶ 843b8.. x2 x4 x5 x6 x7 x8 x9 x10 ⟶ ∀ x11 : ο . (not (x2 x4 x3) ⟶ not (x2 x5 x3) ⟶ not (x2 x6 x3) ⟶ not (x2 x7 x3) ⟶ not (x2 x8 x3) ⟶ not (x2 x9 x3) ⟶ not (x2 x10 x3) ⟶ x11) ⟶ (x2 x4 x3 ⟶ not (x2 x5 x3) ⟶ not (x2 x6 x3) ⟶ not (x2 x7 x3) ⟶ not (x2 x8 x3) ⟶ not (x2 x9 x3) ⟶ not (x2 x10 x3) ⟶ x11) ⟶ (not (x2 x4 x3) ⟶ x2 x5 x3 ⟶ not (x2 x6 x3) ⟶ not (x2 x7 x3) ⟶ not (x2 x8 x3) ⟶ not (x2 x9 x3) ⟶ not (x2 x10 x3) ⟶ x11) ⟶ (x2 x4 x3 ⟶ x2 x5 x3 ⟶ not (x2 x6 x3) ⟶ not (x2 x7 x3) ⟶ not (x2 x8 x3) ⟶ not (x2 x9 x3) ⟶ not (x2 x10 x3) ⟶ x11) ⟶ (not (x2 x4 x3) ⟶ not (x2 x5 x3) ⟶ x2 x6 x3 ⟶ not (x2 x7 x3) ⟶ not (x2 x8 x3) ⟶ not (x2 x9 x3) ⟶ not (x2 x10 x3) ⟶ x11) ⟶ (x2 x4 x3 ⟶ not (x2 x5 x3) ⟶ x2 x6 x3 ⟶ not (x2 x7 x3) ⟶ not (x2 x8 x3) ⟶ not (x2 x9 x3) ⟶ not (x2 x10 x3) ⟶ x11) ⟶ (not (x2 x4 x3) ⟶ x2 x5 x3 ⟶ x2 x6 x3 ⟶ not (x2 x7 x3) ⟶ not (x2 x8 x3) ⟶ not (x2 x9 x3) ⟶ not (x2 x10 x3) ⟶ x11) ⟶ (x2 x4 x3 ⟶ x2 x5 x3 ⟶ x2 x6 x3 ⟶ not (x2 x7 x3) ⟶ not (x2 x8 x3) ⟶ not (x2 x9 x3) ⟶ not (x2 x10 x3) ⟶ x11) ⟶ (not (x2 x4 x3) ⟶ not (x2 x5 x3) ⟶ not (x2 x6 x3) ⟶ x2 x7 x3 ⟶ not (x2 x8 x3) ⟶ not (x2 x9 x3) ⟶ not (x2 x10 x3) ⟶ x11) ⟶ (x2 x4 x3 ⟶ not (x2 x5 x3) ⟶ not (x2 x6 x3) ⟶ x2 x7 x3 ⟶ not (x2 x8 x3) ⟶ not (x2 x9 x3) ⟶ not (x2 x10 x3) ⟶ x11) ⟶ (not (x2 x4 x3) ⟶ not (x2 x5 x3) ⟶ not (x2 x6 x3) ⟶ not (x2 x7 x3) ⟶ x2 x8 x3 ⟶ not (x2 x9 x3) ⟶ not (x2 x10 x3) ⟶ x11) ⟶ (x2 x4 x3 ⟶ not (x2 x5 x3) ⟶ not (x2 x6 x3) ⟶ not (x2 x7 x3) ⟶ x2 x8 x3 ⟶ not (x2 x9 x3) ⟶ not (x2 x10 x3) ⟶ x11) ⟶ (not (x2 x4 x3) ⟶ not (x2 x5 x3) ⟶ not (x2 x6 x3) ⟶ x2 x7 x3 ⟶ x2 x8 x3 ⟶ not (x2 x9 x3) ⟶ not (x2 x10 x3) ⟶ x11) ⟶ (x2 x4 x3 ⟶ not (x2 x5 x3) ⟶ not (x2 x6 x3) ⟶ x2 x7 x3 ⟶ x2 x8 x3 ⟶ not (x2 x9 x3) ⟶ not (x2 x10 x3) ⟶ x11) ⟶ (not (x2 x4 x3) ⟶ not (x2 x5 x3) ⟶ not (x2 x6 x3) ⟶ not (x2 x7 x3) ⟶ not (x2 x8 x3) ⟶ x2 x9 x3 ⟶ not (x2 x10 x3) ⟶ x11) ⟶ (not (x2 x4 x3) ⟶ x2 x5 x3 ⟶ not (x2 x6 x3) ⟶ not (x2 x7 x3) ⟶ not (x2 x8 x3) ⟶ x2 x9 x3 ⟶ not (x2 x10 x3) ⟶ x11) ⟶ (not (x2 x4 x3) ⟶ not (x2 x5 x3) ⟶ x2 x6 x3 ⟶ not (x2 x7 x3) ⟶ not (x2 x8 x3) ⟶ x2 x9 x3 ⟶ not (x2 x10 x3) ⟶ x11) ⟶ (not (x2 x4 x3) ⟶ x2 x5 x3 ⟶ x2 x6 x3 ⟶ not (x2 x7 x3) ⟶ not (x2 x8 x3) ⟶ x2 x9 x3 ⟶ not (x2 x10 x3) ⟶ x11) ⟶ (not (x2 x4 x3) ⟶ not (x2 x5 x3) ⟶ not (x2 x6 x3) ⟶ not (x2 x7 x3) ⟶ not (x2 x8 x3) ⟶ not (x2 x9 x3) ⟶ x2 x10 x3 ⟶ x11) ⟶ (not (x2 x4 x3) ⟶ not (x2 x5 x3) ⟶ not (x2 x6 x3) ⟶ x2 x7 x3 ⟶ not (x2 x8 x3) ⟶ not (x2 x9 x3) ⟶ x2 x10 x3 ⟶ x11) ⟶ (not (x2 x4 x3) ⟶ not (x2 x5 x3) ⟶ not (x2 x6 x3) ⟶ not (x2 x7 x3) ⟶ x2 x8 x3 ⟶ not (x2 x9 x3) ⟶ x2 x10 x3 ⟶ x11) ⟶ (not (x2 x4 x3) ⟶ not (x2 x5 x3) ⟶ not (x2 x6 x3) ⟶ x2 x7 x3 ⟶ x2 x8 x3 ⟶ not (x2 x9 x3) ⟶ x2 x10 x3 ⟶ x11) ⟶ (not (x2 x4 x3) ⟶ not (x2 x5 x3) ⟶ not (x2 x6 x3) ⟶ not (x2 x7 x3) ⟶ not (x2 x8 x3) ⟶ x2 x9 x3 ⟶ x2 x10 x3 ⟶ x11) ⟶ x11 (proof)