Proofgold Address

Definition Subq^Subq := λ x0 x1 . ∀ x2 . x2 ∈ x0 ⟶ x2 ∈ x1
Definition and^and := λ x0 x1 : ο . ∀ x2 : ο . (x0 ⟶ x1 ⟶ x2) ⟶ x2
Definition inj^inj := λ x0 x1 . λ x2 : ι → ι . and (∀ x3 . x3 ∈ x0 ⟶ x2 x3 ∈ x1) (∀ x3 . x3 ∈ x0 ⟶ ∀ x4 . x4 ∈ x0 ⟶ x2 x3 = x2 x4 ⟶ x3 = x4)
Definition atleastp^atleastp := λ x0 x1 . ∀ x2 : ο . (∀ x3 : ι → ι . inj x0 x1 x3 ⟶ x2) ⟶ x2
Param u3 : ι
Definition False^False := ∀ x0 : ο . x0
Definition not^not := λ x0 : ο . x0 ⟶ False
Param nat_p^nat_p : ι → ο
Param setminus^setminus : ι → ι → ι
Param Sep^Sep : ι → (ι → ο) → ι
Definition DirGraphOutNeighbors := λ x0 . λ x1 : ι → ι → ο . λ x2 . {x3 ∈ x0|and (x2 = x3 ⟶ ∀ x4 : ο . x4) (x1 x2 x3)}
Param Sing^Sing : ι → ι
Definition nIn^nIn := λ x0 x1 . not (x0 ∈ x1)
Param binunion^binunion : ι → ι → ι
Param equip^equip : ι → ι → ο
Param binintersect^binintersect : ι → ι → ι
Known atleastp_tra^atleastp_tra : ∀ x0 x1 x2 . atleastp x0 x1 ⟶ atleastp x1 x2 ⟶ atleastp x0 x2
Param inv^inv : ι → (ι → ι) → ι → ι
Known andI^andI : ∀ x0 x1 : ο . x0 ⟶ x1 ⟶ and x0 x1
Known surj_rinv^surj_rinv : ∀ x0 x1 . ∀ x2 : ι → ι . (∀ x3 . x3 ∈ x1 ⟶ ∀ x4 : ο . (∀ x5 . and (x5 ∈ x0) (x2 x5 = x3) ⟶ x4) ⟶ x4) ⟶ ∀ x3 . x3 ∈ x1 ⟶ and (inv x0 x2 x3 ∈ x0) (x2 (inv x0 x2 x3) = x3)
Known binintersectE^{binintersectE} : ∀ x0 x1 x2 . x2 ∈ binintersect x0 x1 ⟶ and (x2 ∈ x0) (x2 ∈ x1)
Known SepI^SepI : ∀ x0 . ∀ x1 : ι → ο . ∀ x2 . x2 ∈ x0 ⟶ x1 x2 ⟶ x2 ∈ Sep x0 x1
Known setminusI^setminusI : ∀ x0 x1 x2 . x2 ∈ x0 ⟶ nIn x2 x1 ⟶ x2 ∈ setminus x0 x1
Known SingE^SingE : ∀ x0 x1 . x1 ∈ Sing x0 ⟶ x1 = x0
Known SepE2^SepE2 : ∀ x0 . ∀ x1 : ι → ο . ∀ x2 . x2 ∈ Sep x0 x1 ⟶ x1 x2
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 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 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 binintersect_com^{binintersect_com} : ∀ x0 x1 . binintersect x0 x1 = binintersect x1 x0
Param SetAdjoin^SetAdjoin : ι → ι → ι
Param UPair^UPair : ι → ι → ι
Known aa241.. : ∀ x0 x1 x2 . ∀ x3 : ι → ο . x3 x0 ⟶ x3 x1 ⟶ x3 x2 ⟶ ∀ x4 . x4 ∈ SetAdjoin (UPair x0 x1) x2 ⟶ x3 x4
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 FalseE^FalseE : False ⟶ ∀ x0 : ο . x0
Theorem 0bee3.. : ∀ 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 x3 x4 . nat_p x4 ⟶ ∀ x5 x6 x7 . x5 ⊆ x0 ⟶ x6 ⊆ x0 ⟶ x7 ⊆ x0 ⟶ x5 = setminus (DirGraphOutNeighbors x0 x1 x3) (Sing x2) ⟶ (∀ x8 . x8 ∈ x6 ⟶ nIn x8 x5) ⟶ (∀ x8 . x8 ∈ x6 ⟶ nIn x8 x7) ⟶ (∀ x8 . x8 ∈ x5 ⟶ nIn x8 x7) ⟶ (∀ x8 . x8 ∈ x7 ⟶ x8 ∈ {x9 ∈ setminus x0 (binunion (DirGraphOutNeighbors x0 x1 x3) (Sing x3))|equip (binintersect (DirGraphOutNeighbors x0 x1 x9) (DirGraphOutNeighbors x0 x1 x3)) x4}) ⟶ ∀ x8 x9 . x8 ∈ x6 ⟶ x9 ∈ x7 ⟶ x9 ∈ setminus x0 (binunion (DirGraphOutNeighbors x0 x1 x2) (Sing x2)) ⟶ x1 x8 x9 ⟶ ∀ x10 : ι → ι . (∀ x11 . x11 ∈ x6 ⟶ x10 x11 ∈ x5) ⟶ (∀ x11 . x11 ∈ x6 ⟶ x1 x11 (x10 x11)) ⟶ (∀ x11 . x11 ∈ x5 ⟶ ∀ x12 : ο . (∀ x13 . and (x13 ∈ x6) (x10 x13 = x11) ⟶ x12) ⟶ x12) ⟶ atleastp x4 {x11 ∈ setminus x6 (Sing x8)|x1 (x10 x11) x9} (proof)
Param u6 : ι
Theorem 867ba.. : ∀ 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 . nat_p x4 ⟶ ∀ x5 x6 x7 . x5 ⊆ x0 ⟶ x6 ⊆ x0 ⟶ x7 ⊆ x0 ⟶ x5 = setminus (DirGraphOutNeighbors x0 x1 x3) (Sing x2) ⟶ (∀ x8 . x8 ∈ x6 ⟶ nIn x8 x5) ⟶ (∀ x8 . x8 ∈ x6 ⟶ nIn x8 x7) ⟶ (∀ x8 . x8 ∈ x5 ⟶ nIn x8 x7) ⟶ (∀ x8 . x8 ∈ x7 ⟶ x8 ∈ {x9 ∈ setminus x0 (binunion (DirGraphOutNeighbors x0 x1 x3) (Sing x3))|equip (binintersect (DirGraphOutNeighbors x0 x1 x9) (DirGraphOutNeighbors x0 x1 x3)) x4}) ⟶ ∀ x8 x9 . x8 ∈ x6 ⟶ x9 ∈ x7 ⟶ x9 ∈ setminus x0 (binunion (DirGraphOutNeighbors x0 x1 x2) (Sing x2)) ⟶ x1 x8 x9 ⟶ ∀ x10 : ι → ι . (∀ x11 . x11 ∈ x6 ⟶ x10 x11 ∈ x5) ⟶ (∀ x11 . x11 ∈ x6 ⟶ x1 x11 (x10 x11)) ⟶ (∀ x11 . x11 ∈ x5 ⟶ ∀ x12 : ο . (∀ x13 . and (x13 ∈ x6) (x10 x13 = x11) ⟶ x12) ⟶ x12) ⟶ atleastp x4 {x11 ∈ setminus x6 (Sing x8)|x1 (x10 x11) x9} (proof)