Proofgold Address

Param equip^equip : ι → ι → ο
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
Definition u1 := 1
Definition u2 := ordsucc u1
Definition u3 := ordsucc u2
Definition u4 := ordsucc u3
Definition u5 := ordsucc u4
Definition Subq^Subq := λ x0 x1 . ∀ x2 . x2 ∈ x0 ⟶ x2 ∈ x1
Param setminus^setminus : ι → ι → ι
Param binunion^binunion : ι → ι → ι
Param Sing^Sing : ι → ι
Param binintersect^binintersect : ι → ι → ι
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
Known dneg^dneg : ∀ x0 : ο . not (not x0) ⟶ x0
Param nat_p^nat_p : ι → ο
Param atleastp^atleastp : ι → ι → ο
Known 4fb58..^{Pigeonhole_not_atleastp_ordsucc} : ∀ x0 . nat_p x0 ⟶ not (atleastp (ordsucc x0) x0)
Known nat_4^nat_4 : nat_p 4
Known atleastp_tra^atleastp_tra : ∀ x0 x1 x2 . atleastp x0 x1 ⟶ atleastp x1 x2 ⟶ atleastp x0 x2
Known equip_atleastp^{equip_atleastp} : ∀ x0 x1 . equip x0 x1 ⟶ atleastp x0 x1
Known equip_sym^equip_sym : ∀ x0 x1 . equip x0 x1 ⟶ equip x1 x0
Known Subq_atleastp^{Subq_atleastp} : ∀ x0 x1 . x0 ⊆ x1 ⟶ atleastp x0 x1
Known SepE^SepE : ∀ x0 . ∀ x1 : ι → ο . ∀ x2 . x2 ∈ Sep x0 x1 ⟶ and (x2 ∈ x0) (x1 x2)
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 binunionI1^binunionI1 : ∀ x0 x1 x2 . x2 ∈ x0 ⟶ x2 ∈ binunion 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 14338.. : ∀ x0 x1 x2 x3 . x2 ∈ SetAdjoin (SetAdjoin (UPair x0 x1) x2) x3
Known setminusE1^setminusE1 : ∀ x0 x1 x2 . x2 ∈ setminus x0 x1 ⟶ x2 ∈ x0
Definition or^or := λ x0 x1 : ο . ∀ x2 : ο . (x0 ⟶ x2) ⟶ (x1 ⟶ x2) ⟶ x2
Known xm^xm : ∀ x0 : ο . or x0 (not x0)
Known b253c.. : ∀ x0 x1 x2 x3 . x3 ∈ SetAdjoin (SetAdjoin (UPair x0 x1) x2) x3
Known binintersectI^{binintersectI} : ∀ x0 x1 x2 . x2 ∈ x0 ⟶ x2 ∈ x1 ⟶ x2 ∈ binintersect x0 x1
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 69a9c.. : ∀ x0 x1 x2 x3 . x0 ∈ SetAdjoin (SetAdjoin (UPair x0 x1) x2) x3
Known e588e.. : ∀ x0 x1 x2 x3 . x1 ∈ SetAdjoin (SetAdjoin (UPair x0 x1) x2) x3
Known 76c0f.. : ∀ x0 x1 x2 x3 . x3 ∈ SetAdjoin (UPair x0 x1) x2 ⟶ ∀ x4 : ι → ο . x4 x0 ⟶ x4 x1 ⟶ x4 x2 ⟶ x4 x3
Known 38089.. : ∀ x0 x1 x2 x3 . atleastp (SetAdjoin (SetAdjoin (UPair x0 x1) x2) x3) u4
Theorem 02907.. : ∀ x0 . ∀ x1 : ι → ι → ο . (∀ x2 x3 . x1 x2 x3 ⟶ x1 x3 x2) ⟶ (∀ x2 . x2 ∈ x0 ⟶ equip (DirGraphOutNeighbors x0 x1 x2) u5) ⟶ ∀ x2 x3 x4 x5 x6 x7 . x6 ⊆ x0 ⟶ x5 ⊆ x0 ⟶ x6 = {x9 ∈ setminus x0 (binunion (DirGraphOutNeighbors x0 x1 x2) (Sing x2))|equip (binintersect (DirGraphOutNeighbors x0 x1 x9) (DirGraphOutNeighbors x0 x1 x2)) u1} ⟶ x4 = setminus (DirGraphOutNeighbors x0 x1 x2) (Sing x3) ⟶ (∀ x8 . x8 ∈ x6 ⟶ not (x1 x3 x8)) ⟶ (∀ x8 . x8 ∈ x0 ⟶ ∀ x9 : ο . (x8 = x2 ⟶ x9) ⟶ (x8 = x3 ⟶ x9) ⟶ (x8 ∈ x4 ⟶ x9) ⟶ (x8 ∈ x5 ⟶ x9) ⟶ (x8 ∈ x6 ⟶ x9) ⟶ (x8 ∈ x7 ⟶ x9) ⟶ x9) ⟶ (∀ x8 . x8 ∈ x6 ⟶ ∀ x9 : ο . (∀ x10 . x10 ∈ x6 ⟶ ∀ x11 . x11 ∈ x6 ⟶ (x10 = x11 ⟶ ∀ x12 : ο . x12) ⟶ x10 ∈ DirGraphOutNeighbors x0 x1 x8 ⟶ x11 ∈ DirGraphOutNeighbors x0 x1 x8 ⟶ not (x1 x10 x11) ⟶ (∀ x12 . x12 ∈ x6 ⟶ nIn x12 (SetAdjoin (UPair x8 x10) x11) ⟶ not (x1 x8 x12)) ⟶ x9) ⟶ x9) ⟶ (∀ x8 . x8 ∈ x6 ⟶ ∀ x9 . x9 ∈ x6 ⟶ (x8 = x9 ⟶ ∀ x10 : ο . x10) ⟶ ∀ x10 . x10 ∈ binintersect (DirGraphOutNeighbors x0 x1 x8) (DirGraphOutNeighbors x0 x1 x9) ⟶ x10 ∈ x6) ⟶ (∀ x8 . x8 ∈ x6 ⟶ nIn x8 x5) ⟶ ∀ x8 x9 : ι → ι . (∀ x10 . x10 ∈ x6 ⟶ ∀ x11 . x11 ∈ DirGraphOutNeighbors x0 x1 x2 ⟶ x1 x11 x10 ⟶ x11 = x8 x10) ⟶ (∀ x10 . x10 ∈ x6 ⟶ x9 x10 ∈ x5) ⟶ (∀ x10 . x10 ∈ x6 ⟶ x1 x10 (x9 x10)) ⟶ (∀ x10 . x10 ∈ x5 ⟶ ∀ x11 : ο . (∀ x12 . and (x12 ∈ x6) (x9 x12 = x10) ⟶ x11) ⟶ x11) ⟶ ∀ x10 . x10 ∈ x6 ⟶ ∀ x11 : ο . (∀ x12 . and (x12 ∈ x7) (x1 x10 x12) ⟶ x11) ⟶ x11 (proof)
Theorem 02907.. : ∀ x0 . ∀ x1 : ι → ι → ο . (∀ x2 x3 . x1 x2 x3 ⟶ x1 x3 x2) ⟶ (∀ x2 . x2 ∈ x0 ⟶ equip (DirGraphOutNeighbors x0 x1 x2) u5) ⟶ ∀ x2 x3 x4 x5 x6 x7 . x6 ⊆ x0 ⟶ x5 ⊆ x0 ⟶ x6 = {x9 ∈ setminus x0 (binunion (DirGraphOutNeighbors x0 x1 x2) (Sing x2))|equip (binintersect (DirGraphOutNeighbors x0 x1 x9) (DirGraphOutNeighbors x0 x1 x2)) u1} ⟶ x4 = setminus (DirGraphOutNeighbors x0 x1 x2) (Sing x3) ⟶ (∀ x8 . x8 ∈ x6 ⟶ not (x1 x3 x8)) ⟶ (∀ x8 . x8 ∈ x0 ⟶ ∀ x9 : ο . (x8 = x2 ⟶ x9) ⟶ (x8 = x3 ⟶ x9) ⟶ (x8 ∈ x4 ⟶ x9) ⟶ (x8 ∈ x5 ⟶ x9) ⟶ (x8 ∈ x6 ⟶ x9) ⟶ (x8 ∈ x7 ⟶ x9) ⟶ x9) ⟶ (∀ x8 . x8 ∈ x6 ⟶ ∀ x9 : ο . (∀ x10 . x10 ∈ x6 ⟶ ∀ x11 . x11 ∈ x6 ⟶ (x10 = x11 ⟶ ∀ x12 : ο . x12) ⟶ x10 ∈ DirGraphOutNeighbors x0 x1 x8 ⟶ x11 ∈ DirGraphOutNeighbors x0 x1 x8 ⟶ not (x1 x10 x11) ⟶ (∀ x12 . x12 ∈ x6 ⟶ nIn x12 (SetAdjoin (UPair x8 x10) x11) ⟶ not (x1 x8 x12)) ⟶ x9) ⟶ x9) ⟶ (∀ x8 . x8 ∈ x6 ⟶ ∀ x9 . x9 ∈ x6 ⟶ (x8 = x9 ⟶ ∀ x10 : ο . x10) ⟶ ∀ x10 . x10 ∈ binintersect (DirGraphOutNeighbors x0 x1 x8) (DirGraphOutNeighbors x0 x1 x9) ⟶ x10 ∈ x6) ⟶ (∀ x8 . x8 ∈ x6 ⟶ nIn x8 x5) ⟶ ∀ x8 x9 : ι → ι . (∀ x10 . x10 ∈ x6 ⟶ ∀ x11 . x11 ∈ DirGraphOutNeighbors x0 x1 x2 ⟶ x1 x11 x10 ⟶ x11 = x8 x10) ⟶ (∀ x10 . x10 ∈ x6 ⟶ x9 x10 ∈ x5) ⟶ (∀ x10 . x10 ∈ x6 ⟶ x1 x10 (x9 x10)) ⟶ (∀ x10 . x10 ∈ x5 ⟶ ∀ x11 : ο . (∀ x12 . and (x12 ∈ x6) (x9 x12 = x10) ⟶ x11) ⟶ x11) ⟶ ∀ x10 . x10 ∈ x6 ⟶ ∀ x11 : ο . (∀ x12 . and (x12 ∈ x7) (x1 x10 x12) ⟶ x11) ⟶ x11 (proof)