Proofgold Address

Definition Subq^Subq := λ x0 x1 . ∀ x2 . x2 ∈ x0 ⟶ x2 ∈ x1
Param atleastp^atleastp : ι → ι → ο
Param ordsucc^ordsucc : ι → ι
Definition u1 := 1
Definition u2 := ordsucc u1
Definition u3 := ordsucc u2
Definition False^False := ∀ x0 : ο . x0
Definition not^not := λ x0 : ο . x0 ⟶ False
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)}
Definition nIn^nIn := λ x0 x1 . not (x0 ∈ x1)
Param SetAdjoin^SetAdjoin : ι → ι → ι
Param UPair^UPair : ι → 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
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
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 6be8c.. : ∀ x0 x1 x2 . x0 ∈ SetAdjoin (UPair x0 x1) x2
Known 535ce.. : ∀ x0 x1 x2 . x1 ∈ SetAdjoin (UPair x0 x1) x2
Known f4e2f.. : ∀ x0 x1 x2 . x2 ∈ SetAdjoin (UPair x0 x1) x2
Known Subq_atleastp^{Subq_atleastp} : ∀ x0 x1 . x0 ⊆ x1 ⟶ atleastp x0 x1
Known aa241.. : ∀ x0 x1 x2 . ∀ x3 : ι → ο . x3 x0 ⟶ x3 x1 ⟶ x3 x2 ⟶ ∀ x4 . x4 ∈ SetAdjoin (UPair x0 x1) x2 ⟶ x3 x4
Known binintersectI^{binintersectI} : ∀ x0 x1 x2 . x2 ∈ x0 ⟶ x2 ∈ x1 ⟶ x2 ∈ binintersect x0 x1
Known cfabd.. : ∀ x0 . ∀ x1 : ι → ι → ο . (∀ x2 x3 . x1 x2 x3 ⟶ x1 x3 x2) ⟶ ∀ x2 . x2 ∈ x0 ⟶ ∀ x3 . x3 ∈ DirGraphOutNeighbors x0 x1 x2 ⟶ x2 ∈ DirGraphOutNeighbors x0 x1 x3
Known b253c.. : ∀ x0 x1 x2 x3 . x3 ∈ SetAdjoin (SetAdjoin (UPair x0 x1) x2) x3
Known 14338.. : ∀ x0 x1 x2 x3 . x2 ∈ 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
Known FalseE^FalseE : False ⟶ ∀ x0 : ο . x0
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
Theorem f0ba0.. : ∀ 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 ∈ x0 ⟶ (x2 = x3 ⟶ ∀ x4 : ο . x4) ⟶ not (x1 x2 x3) ⟶ atleastp (binintersect (DirGraphOutNeighbors x0 x1 x2) (DirGraphOutNeighbors x0 x1 x3)) u2) ⟶ ∀ x2 x3 x4 x5 . x2 ⊆ x0 ⟶ x3 ⊆ x0 ⟶ x4 ⊆ x0 ⟶ x5 ⊆ x0 ⟶ (∀ x6 . x6 ∈ x4 ⟶ nIn x6 x2) ⟶ (∀ x6 . x6 ∈ x4 ⟶ nIn x6 x3) ⟶ (∀ x6 . x6 ∈ x4 ⟶ nIn x6 x5) ⟶ (∀ x6 . x6 ∈ x2 ⟶ nIn x6 x3) ⟶ (∀ x6 . x6 ∈ x2 ⟶ nIn x6 x5) ⟶ (∀ x6 . x6 ∈ x3 ⟶ nIn x6 x5) ⟶ ∀ x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 . x6 ∈ x4 ⟶ x7 ∈ x4 ⟶ x3 = SetAdjoin (SetAdjoin (UPair x10 x11) x12) x13 ⟶ x14 ∈ x2 ⟶ x15 ∈ x5 ⟶ (x6 = x7 ⟶ ∀ x16 : ο . x16) ⟶ x1 x6 x7 ⟶ x1 x6 x10 ⟶ x1 x6 x15 ⟶ x1 x7 x14 ⟶ x1 x7 x11 ⟶ x1 x12 x15 ⟶ not (x1 x14 x6) ⟶ x1 x14 x15 ⟶ x1 x13 x15 ⟶ ∀ x16 . x16 ∈ x3 ⟶ not (x1 x14 x16) (proof)
Param u6 : ι
Known neq_i_sym^neq_i_sym : ∀ x0 x1 . (x0 = x1 ⟶ ∀ x2 : ο . x2) ⟶ x1 = x0 ⟶ ∀ x2 : ο . x2
Theorem 50435.. : ∀ 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 . x2 ∈ x0 ⟶ ∀ x3 . x3 ∈ x0 ⟶ (x2 = x3 ⟶ ∀ x4 : ο . x4) ⟶ not (x1 x2 x3) ⟶ atleastp (binintersect (DirGraphOutNeighbors x0 x1 x2) (DirGraphOutNeighbors x0 x1 x3)) u2) ⟶ ∀ x2 x3 x4 x5 . x2 ⊆ x0 ⟶ x3 ⊆ x0 ⟶ x4 ⊆ x0 ⟶ x5 ⊆ x0 ⟶ (∀ x6 . x6 ∈ x4 ⟶ nIn x6 x2) ⟶ (∀ x6 . x6 ∈ x4 ⟶ nIn x6 x3) ⟶ (∀ x6 . x6 ∈ x4 ⟶ nIn x6 x5) ⟶ (∀ x6 . x6 ∈ x2 ⟶ nIn x6 x3) ⟶ (∀ x6 . x6 ∈ x2 ⟶ nIn x6 x5) ⟶ (∀ x6 . x6 ∈ x3 ⟶ nIn x6 x5) ⟶ ∀ x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 . x6 ∈ x4 ⟶ x7 ∈ x4 ⟶ x3 = SetAdjoin (SetAdjoin (UPair x10 x11) x12) x13 ⟶ x14 ∈ x2 ⟶ x15 ∈ x5 ⟶ (x7 = x6 ⟶ ∀ x16 : ο . x16) ⟶ (x8 = x6 ⟶ ∀ x16 : ο . x16) ⟶ (x9 = x6 ⟶ ∀ x16 : ο . x16) ⟶ (x8 = x7 ⟶ ∀ x16 : ο . x16) ⟶ (x9 = x7 ⟶ ∀ x16 : ο . x16) ⟶ (x9 = x8 ⟶ ∀ x16 : ο . x16) ⟶ x1 x6 x7 ⟶ x1 x7 x8 ⟶ x1 x8 x9 ⟶ x1 x9 x6 ⟶ x1 x6 x10 ⟶ x1 x6 x15 ⟶ x1 x7 x14 ⟶ x1 x7 x11 ⟶ x1 x12 x15 ⟶ not (x1 x14 x6) ⟶ x1 x14 x15 ⟶ x1 x13 x15 ⟶ ∀ x16 . x16 ∈ x3 ⟶ not (x1 x14 x16) (proof)