Proofgold Address

Definition Subq^Subq := λ x0 x1 . ∀ x2 . x2 ∈ x0 ⟶ x2 ∈ x1
Definition False^False := ∀ x0 : ο . x0
Definition not^not := λ x0 : ο . x0 ⟶ False
Definition nIn^nIn := λ x0 x1 . not (x0 ∈ x1)
Param SetAdjoin^SetAdjoin : ι → ι → ι
Param UPair^UPair : ι → ι → ι
Param atleastp^atleastp : ι → ι → ο
Param binintersect^binintersect : ι → ι → ι
Param Sep^Sep : ι → (ι → ο) → ι
Param and^and : ο → ο → ο
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
Definition u1 := 1
Definition u2 := ordsucc u1
Param setminus^setminus : ι → ι → ι
Param Sing^Sing : ι → ι
Known 8698a.. : ∀ x0 x1 x2 x3 . ∀ x4 : ι → ο . x4 x0 ⟶ x4 x1 ⟶ x4 x2 ⟶ x4 x3 ⟶ ∀ x5 . x5 ∈ SetAdjoin (SetAdjoin (UPair x0 x1) x2) x3 ⟶ x4 x5
Known FalseE^FalseE : False ⟶ ∀ x0 : ο . x0
Known setminusE2^setminusE2 : ∀ x0 x1 x2 . x2 ∈ setminus x0 x1 ⟶ nIn x2 x1
Known SepE1^SepE1 : ∀ x0 . ∀ x1 : ι → ο . ∀ x2 . x2 ∈ Sep x0 x1 ⟶ x2 ∈ x0
Known SingI^SingI : ∀ x0 . x0 ∈ Sing x0
Param nat_p^nat_p : ι → ο
Known 4fb58..^{Pigeonhole_not_atleastp_ordsucc} : ∀ x0 . nat_p x0 ⟶ not (atleastp (ordsucc x0) x0)
Known nat_2^nat_2 : nat_p 2
Definition u3 := ordsucc u2
Known atleastp_tra^atleastp_tra : ∀ x0 x1 x2 . atleastp x0 x1 ⟶ atleastp x1 x2 ⟶ atleastp x0 x2
Known 5d098.. : ∀ x0 x1 . x1 ∈ x0 ⟶ ∀ x2 . x2 ∈ x0 ⟶ ∀ x3 . x3 ∈ x0 ⟶ (x1 = x2 ⟶ ∀ x4 : ο . x4) ⟶ (x1 = x3 ⟶ ∀ x4 : ο . x4) ⟶ (x2 = x3 ⟶ ∀ x4 : ο . x4) ⟶ atleastp u3 x0
Known binintersectI^{binintersectI} : ∀ x0 x1 x2 . x2 ∈ x0 ⟶ x2 ∈ x1 ⟶ x2 ∈ binintersect x0 x1
Known SepE2^SepE2 : ∀ x0 . ∀ x1 : ι → ο . ∀ x2 . x2 ∈ Sep x0 x1 ⟶ x1 x2
Known neq_i_sym^neq_i_sym : ∀ x0 x1 . (x0 = x1 ⟶ ∀ x2 : ο . x2) ⟶ x1 = x0 ⟶ ∀ x2 : ο . x2
Known SepI^SepI : ∀ x0 . ∀ x1 : ι → ο . ∀ x2 . x2 ∈ x0 ⟶ x1 x2 ⟶ x2 ∈ Sep x0 x1
Known andI^andI : ∀ x0 x1 : ο . x0 ⟶ x1 ⟶ and x0 x1
Known b253c.. : ∀ x0 x1 x2 x3 . x3 ∈ SetAdjoin (SetAdjoin (UPair x0 x1) x2) x3
Known e588e.. : ∀ x0 x1 x2 x3 . x1 ∈ SetAdjoin (SetAdjoin (UPair x0 x1) x2) x3
Known 69a9c.. : ∀ x0 x1 x2 x3 . x0 ∈ SetAdjoin (SetAdjoin (UPair x0 x1) x2) x3
Theorem 3eb85.. : ∀ x0 . ∀ x1 : ι → ι → ο . (∀ x2 x3 . x1 x2 x3 ⟶ x1 x3 x2) ⟶ ∀ x2 x3 x4 x5 . x2 ⊆ x0 ⟶ x3 ⊆ x0 ⟶ x4 ⊆ x0 ⟶ x5 ⊆ x0 ⟶ (∀ x6 . x6 ∈ x2 ⟶ nIn x6 x5) ⟶ (∀ x6 . x6 ∈ x2 ⟶ nIn x6 x3) ⟶ (∀ x6 . x6 ∈ x4 ⟶ nIn x6 x2) ⟶ (∀ x6 . x6 ∈ x4 ⟶ nIn x6 x3) ⟶ (∀ x6 . x6 ∈ x4 ⟶ nIn x6 x5) ⟶ (∀ x6 . x6 ∈ x3 ⟶ nIn x6 x5) ⟶ ∀ x6 x7 x8 x9 x10 . x4 = SetAdjoin (SetAdjoin (UPair x6 x7) x8) x9 ⟶ x10 ∈ x5 ⟶ (∀ x11 . x11 ∈ x4 ⟶ (x11 = x10 ⟶ ∀ x12 : ο . x12) ⟶ not (x1 x11 x10) ⟶ atleastp (binintersect (DirGraphOutNeighbors x0 x1 x11) (DirGraphOutNeighbors x0 x1 x10)) u2) ⟶ x6 ∈ binintersect (DirGraphOutNeighbors x0 x1 x7) (DirGraphOutNeighbors x0 x1 x10) ⟶ x6 ∈ binintersect (DirGraphOutNeighbors x0 x1 x9) (DirGraphOutNeighbors x0 x1 x10) ⟶ not (x1 x7 x10) ⟶ not (x1 x9 x10) ⟶ ∀ x11 x12 : ι → ι . (∀ x13 . x13 ∈ x4 ⟶ x11 x13 ∈ x2) ⟶ (∀ x13 . x13 ∈ x4 ⟶ x11 x13 ∈ DirGraphOutNeighbors x0 x1 x13) ⟶ (∀ x13 . x13 ∈ x4 ⟶ x12 x13 ∈ x3) ⟶ (∀ x13 . x13 ∈ x4 ⟶ x12 x13 ∈ DirGraphOutNeighbors x0 x1 x13) ⟶ ∀ x13 . x13 ∈ x4 ⟶ x13 ∈ {x14 ∈ setminus x4 (Sing x6)|x1 (x11 x14) x10} ⟶ x13 ∈ {x14 ∈ setminus x4 (Sing x6)|x1 (x12 x14) x10} ⟶ x13 = x8 (proof)
Param u6 : ι
Theorem 8acb5.. : ∀ 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 x3 x4 x5 . x2 ⊆ x0 ⟶ x3 ⊆ x0 ⟶ x4 ⊆ x0 ⟶ x5 ⊆ x0 ⟶ (∀ x6 . x6 ∈ x2 ⟶ nIn x6 x5) ⟶ (∀ x6 . x6 ∈ x2 ⟶ nIn x6 x3) ⟶ (∀ x6 . x6 ∈ x4 ⟶ nIn x6 x2) ⟶ (∀ x6 . x6 ∈ x4 ⟶ nIn x6 x3) ⟶ (∀ x6 . x6 ∈ x4 ⟶ nIn x6 x5) ⟶ (∀ x6 . x6 ∈ x3 ⟶ nIn x6 x5) ⟶ ∀ x6 x7 x8 x9 x10 . x4 = SetAdjoin (SetAdjoin (UPair x6 x7) x8) x9 ⟶ x10 ∈ x5 ⟶ (x7 = x6 ⟶ ∀ x11 : ο . x11) ⟶ (x8 = x6 ⟶ ∀ x11 : ο . x11) ⟶ (x9 = x6 ⟶ ∀ x11 : ο . x11) ⟶ (x8 = x7 ⟶ ∀ x11 : ο . x11) ⟶ (x9 = x7 ⟶ ∀ x11 : ο . x11) ⟶ (x9 = x8 ⟶ ∀ x11 : ο . x11) ⟶ x1 x6 x7 ⟶ x1 x7 x8 ⟶ x1 x8 x9 ⟶ x1 x9 x6 ⟶ (∀ x11 . x11 ∈ x4 ⟶ (x11 = x10 ⟶ ∀ x12 : ο . x12) ⟶ not (x1 x11 x10) ⟶ atleastp (binintersect (DirGraphOutNeighbors x0 x1 x11) (DirGraphOutNeighbors x0 x1 x10)) u2) ⟶ x6 ∈ binintersect (DirGraphOutNeighbors x0 x1 x7) (DirGraphOutNeighbors x0 x1 x10) ⟶ x6 ∈ binintersect (DirGraphOutNeighbors x0 x1 x9) (DirGraphOutNeighbors x0 x1 x10) ⟶ not (x1 x7 x10) ⟶ not (x1 x9 x10) ⟶ ∀ x11 x12 : ι → ι . (∀ x13 . x13 ∈ x4 ⟶ x11 x13 ∈ x2) ⟶ (∀ x13 . x13 ∈ x4 ⟶ x11 x13 ∈ DirGraphOutNeighbors x0 x1 x13) ⟶ (∀ x13 . x13 ∈ x4 ⟶ x12 x13 ∈ x3) ⟶ (∀ x13 . x13 ∈ x4 ⟶ x12 x13 ∈ DirGraphOutNeighbors x0 x1 x13) ⟶ ∀ x13 . x13 ∈ x4 ⟶ x13 ∈ {x14 ∈ setminus x4 (Sing x6)|x1 (x11 x14) x10} ⟶ x13 ∈ {x14 ∈ setminus x4 (Sing x6)|x1 (x12 x14) x10} ⟶ x13 = x8 (proof)