Proofgold Address

Definition Subq^Subq := λ x0 x1 . ∀ x2 . x2 ∈ x0 ⟶ x2 ∈ x1
Param atleastp^atleastp : ι → ι → ο
Param u3 : ι
Definition False^False := ∀ x0 : ο . x0
Definition not^not := λ x0 : ο . x0 ⟶ False
Param nat_p^nat_p : ι → ο
Definition or^or := λ x0 x1 : ο . ∀ x2 : ο . (x0 ⟶ x2) ⟶ (x1 ⟶ x2) ⟶ x2
Param equip^equip : ι → ι → ο
Param binintersect^binintersect : ι → ι → ι
Param Sep^Sep : ι → (ι → ο) → ι
Definition and^and := λ x0 x1 : ο . ∀ x2 : ο . (x0 ⟶ x1 ⟶ x2) ⟶ x2
Definition DirGraphOutNeighbors := λ x0 . λ x1 : ι → ι → ο . λ x2 . {x3 ∈ x0|and (x2 = x3 ⟶ ∀ x4 : ο . x4) (x1 x2 x3)}
Param ordsucc^ordsucc : ι → ι
Param setminus^setminus : ι → ι → ι
Param binunion^binunion : ι → ι → ι
Param Sing^Sing : ι → ι
Known binunion_com^binunion_com : ∀ x0 x1 . binunion x0 x1 = binunion x1 x0
Known e6195.. : ∀ x0 x1 . x1 ⊆ x0 ⟶ x0 = binunion (setminus x0 x1) x1
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)
Known SepE^SepE : ∀ x0 . ∀ x1 : ι → ο . ∀ x2 . x2 ∈ Sep x0 x1 ⟶ and (x2 ∈ x0) (x1 x2)
Known SepI^SepI : ∀ x0 . ∀ x1 : ι → ο . ∀ x2 . x2 ∈ x0 ⟶ x1 x2 ⟶ x2 ∈ Sep x0 x1
Known FalseE^FalseE : False ⟶ ∀ x0 : ο . x0
Known setminusE2^setminusE2 : ∀ x0 x1 x2 . x2 ∈ setminus x0 x1 ⟶ nIn x2 x1
Known binunionI1^binunionI1 : ∀ x0 x1 x2 . x2 ∈ x0 ⟶ x2 ∈ binunion x0 x1
Known andI^andI : ∀ x0 x1 : ο . x0 ⟶ x1 ⟶ and x0 x1
Known neq_i_sym^neq_i_sym : ∀ x0 x1 . (x0 = x1 ⟶ ∀ x2 : ο . x2) ⟶ x1 = x0 ⟶ ∀ x2 : ο . x2
Known SingI^SingI : ∀ x0 . x0 ∈ Sing x0
Known setminusI^setminusI : ∀ x0 x1 x2 . x2 ∈ x0 ⟶ nIn x2 x1 ⟶ x2 ∈ setminus x0 x1
Known binunionE^binunionE : ∀ x0 x1 x2 . x2 ∈ binunion x0 x1 ⟶ or (x2 ∈ x0) (x2 ∈ x1)
Known 7fc90.. : ∀ x0 . ∀ x1 : ι → ι → ο . (∀ x2 x3 . x1 x2 x3 ⟶ x1 x3 x2) ⟶ (∀ x2 . x2 ⊆ x0 ⟶ atleastp u3 x2 ⟶ not (∀ x3 . x3 ∈ x2 ⟶ ∀ x4 . x4 ∈ x2 ⟶ (x3 = x4 ⟶ ∀ x5 : ο . x5) ⟶ x1 x3 x4)) ⟶ ∀ x2 . x2 ∈ x0 ⟶ ∀ x3 . x3 ∈ DirGraphOutNeighbors x0 x1 x2 ⟶ ∀ x4 . x4 ∈ DirGraphOutNeighbors x0 x1 x2 ⟶ (x3 = x4 ⟶ ∀ x5 : ο . x5) ⟶ not (x1 x3 x4)
Theorem 05419.. : ∀ x0 . ∀ x1 : ι → ι → ο . (∀ x2 x3 . x1 x2 x3 ⟶ x1 x3 x2) ⟶ (∀ x2 . x2 ⊆ x0 ⟶ atleastp u3 x2 ⟶ not (∀ x3 . x3 ∈ x2 ⟶ ∀ x4 . x4 ∈ x2 ⟶ (x3 = x4 ⟶ ∀ x5 : ο . x5) ⟶ x1 x3 x4)) ⟶ ∀ x2 . nat_p x2 ⟶ (∀ x3 . x3 ∈ x0 ⟶ ∀ x4 . x4 ∈ x0 ⟶ (x3 = x4 ⟶ ∀ x5 : ο . x5) ⟶ not (x1 x3 x4) ⟶ or (equip (binintersect (DirGraphOutNeighbors x0 x1 x3) (DirGraphOutNeighbors x0 x1 x4)) x2) (equip (binintersect (DirGraphOutNeighbors x0 x1 x3) (DirGraphOutNeighbors x0 x1 x4)) (ordsucc x2))) ⟶ ∀ x3 x4 . x3 ∈ x0 ⟶ x4 ∈ DirGraphOutNeighbors x0 x1 x3 ⟶ (∀ x5 . x5 ∈ {x6 ∈ setminus x0 (binunion (DirGraphOutNeighbors x0 x1 x3) (Sing x3))|equip (binintersect (DirGraphOutNeighbors x0 x1 x6) (DirGraphOutNeighbors x0 x1 x3)) x2} ⟶ not (x1 x4 x5)) ⟶ binunion (setminus (DirGraphOutNeighbors x0 x1 x4) (Sing x3)) (setminus {x6 ∈ setminus x0 (binunion (DirGraphOutNeighbors x0 x1 x3) (Sing x3))|equip (binintersect (DirGraphOutNeighbors x0 x1 x6) (DirGraphOutNeighbors x0 x1 x3)) (ordsucc x2)} (setminus (DirGraphOutNeighbors x0 x1 x4) (Sing x3))) = {x6 ∈ setminus x0 (binunion (DirGraphOutNeighbors x0 x1 x3) (Sing x3))|equip (binintersect (DirGraphOutNeighbors x0 x1 x6) (DirGraphOutNeighbors x0 x1 x3)) (ordsucc x2)} (proof)
Param u6 : ι
Theorem 04590.. : ∀ x0 . ∀ x1 : ι → ι → ο . (∀ x2 x3 . x1 x2 x3 ⟶ x1 x3 x2) ⟶ (∀ x2 . x2 ⊆ x0 ⟶ atleastp u3 x2 ⟶ not (∀ x3 . x3 ∈ x2 ⟶ ∀ x4 . x4 ∈ x2 ⟶ (x3 = x4 ⟶ ∀ x5 : ο . x5) ⟶ x1 x3 x4)) ⟶ (∀ x2 . x2 ⊆ x0 ⟶ atleastp u6 x2 ⟶ not (∀ x3 . x3 ∈ x2 ⟶ ∀ x4 . x4 ∈ x2 ⟶ (x3 = x4 ⟶ ∀ x5 : ο . x5) ⟶ not (x1 x3 x4))) ⟶ ∀ x2 . nat_p x2 ⟶ (∀ x3 . x3 ∈ x0 ⟶ ∀ x4 . x4 ∈ x0 ⟶ (x3 = x4 ⟶ ∀ x5 : ο . x5) ⟶ not (x1 x3 x4) ⟶ or (equip (binintersect (DirGraphOutNeighbors x0 x1 x3) (DirGraphOutNeighbors x0 x1 x4)) x2) (equip (binintersect (DirGraphOutNeighbors x0 x1 x3) (DirGraphOutNeighbors x0 x1 x4)) (ordsucc x2))) ⟶ ∀ x3 x4 . x3 ∈ x0 ⟶ x4 ∈ DirGraphOutNeighbors x0 x1 x3 ⟶ (∀ x5 . x5 ∈ {x6 ∈ setminus x0 (binunion (DirGraphOutNeighbors x0 x1 x3) (Sing x3))|equip (binintersect (DirGraphOutNeighbors x0 x1 x6) (DirGraphOutNeighbors x0 x1 x3)) x2} ⟶ not (x1 x4 x5)) ⟶ binunion (setminus (DirGraphOutNeighbors x0 x1 x4) (Sing x3)) (setminus {x6 ∈ setminus x0 (binunion (DirGraphOutNeighbors x0 x1 x3) (Sing x3))|equip (binintersect (DirGraphOutNeighbors x0 x1 x6) (DirGraphOutNeighbors x0 x1 x3)) (ordsucc x2)} (setminus (DirGraphOutNeighbors x0 x1 x4) (Sing x3))) = {x6 ∈ setminus x0 (binunion (DirGraphOutNeighbors x0 x1 x3) (Sing x3))|equip (binintersect (DirGraphOutNeighbors x0 x1 x6) (DirGraphOutNeighbors x0 x1 x3)) (ordsucc x2)} (proof)