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 SetAdjoin^SetAdjoin : ι → ι → ι
Param UPair^UPair : ι → ι → ι
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)}
Definition or^or := λ x0 x1 : ο . ∀ x2 : ο . (x0 ⟶ x2) ⟶ (x1 ⟶ x2) ⟶ x2
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 binintersectE^{binintersectE} : ∀ x0 x1 x2 . x2 ∈ binintersect x0 x1 ⟶ and (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)
Known neq_i_sym^neq_i_sym : ∀ x0 x1 . (x0 = x1 ⟶ ∀ x2 : ο . x2) ⟶ x1 = x0 ⟶ ∀ x2 : ο . x2
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 SepE1^SepE1 : ∀ x0 . ∀ x1 : ι → ο . ∀ x2 . x2 ∈ Sep x0 x1 ⟶ x2 ∈ x0
Known binintersect_com^{binintersect_com} : ∀ x0 x1 . binintersect x0 x1 = binintersect x1 x0
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
Theorem 782e4.. : ∀ 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 x4 x5 x6 . x2 = SetAdjoin (SetAdjoin (UPair x3 x4) x5) x6 ⟶ (x4 = x3 ⟶ ∀ x7 : ο . x7) ⟶ (x5 = x3 ⟶ ∀ x7 : ο . x7) ⟶ (x6 = x3 ⟶ ∀ x7 : ο . x7) ⟶ (x5 = x4 ⟶ ∀ x7 : ο . x7) ⟶ (x6 = x4 ⟶ ∀ x7 : ο . x7) ⟶ (x6 = x5 ⟶ ∀ x7 : ο . x7) ⟶ x1 x3 x4 ⟶ x1 x4 x5 ⟶ x1 x5 x6 ⟶ x1 x6 x3 ⟶ (∀ x7 . x7 ∈ binintersect (DirGraphOutNeighbors x0 x1 x3) (DirGraphOutNeighbors x0 x1 x5) ⟶ or (x7 = x4) (x7 = x6)) ⟶ (∀ x7 . x7 ∈ binintersect (DirGraphOutNeighbors x0 x1 x4) (DirGraphOutNeighbors x0 x1 x6) ⟶ or (x7 = x3) (x7 = x5)) ⟶ ∀ x7 . x7 ∈ x2 ⟶ ∀ x8 . x8 ∈ x2 ⟶ (x7 = x8 ⟶ ∀ x9 : ο . x9) ⟶ ∀ x9 . x9 ∈ binintersect (DirGraphOutNeighbors x0 x1 x7) (DirGraphOutNeighbors x0 x1 x8) ⟶ x9 ∈ x2 (proof)
Param u6 : ι
Theorem 49b78.. : ∀ 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 x4 x5 x6 . x2 = SetAdjoin (SetAdjoin (UPair x3 x4) x5) x6 ⟶ (x4 = x3 ⟶ ∀ x7 : ο . x7) ⟶ (x5 = x3 ⟶ ∀ x7 : ο . x7) ⟶ (x6 = x3 ⟶ ∀ x7 : ο . x7) ⟶ (x5 = x4 ⟶ ∀ x7 : ο . x7) ⟶ (x6 = x4 ⟶ ∀ x7 : ο . x7) ⟶ (x6 = x5 ⟶ ∀ x7 : ο . x7) ⟶ x1 x3 x4 ⟶ x1 x4 x5 ⟶ x1 x5 x6 ⟶ x1 x6 x3 ⟶ (∀ x7 . x7 ∈ binintersect (DirGraphOutNeighbors x0 x1 x3) (DirGraphOutNeighbors x0 x1 x5) ⟶ or (x7 = x4) (x7 = x6)) ⟶ (∀ x7 . x7 ∈ binintersect (DirGraphOutNeighbors x0 x1 x4) (DirGraphOutNeighbors x0 x1 x6) ⟶ or (x7 = x3) (x7 = x5)) ⟶ ∀ x7 . x7 ∈ x2 ⟶ ∀ x8 . x8 ∈ x2 ⟶ (x7 = x8 ⟶ ∀ x9 : ο . x9) ⟶ ∀ x9 . x9 ∈ binintersect (DirGraphOutNeighbors x0 x1 x7) (DirGraphOutNeighbors x0 x1 x8) ⟶ x9 ∈ x2 (proof)