Proofgold Address

Param equip^equip : ι → ι → ο
Param binunion^binunion : ι → ι → ι
Param Sing^Sing : ι → ι
Definition SetAdjoin^SetAdjoin := λ x0 x1 . binunion x0 (Sing x1)
Param UPair^UPair : ι → ι → ι
Param ordsucc^ordsucc : ι → ι
Definition u1 := 1
Definition u2 := ordsucc u1
Definition u3 := ordsucc u2
Definition u4 := ordsucc u3
Definition u5 := ordsucc u4
Known equip_sym^equip_sym : ∀ x0 x1 . equip x0 x1 ⟶ equip x1 x0
Definition False^False := ∀ x0 : ο . x0
Definition not^not := λ x0 : ο . x0 ⟶ False
Definition nIn^nIn := λ x0 x1 . not (x0 ∈ x1)
Known e8bc0..^{equip_adjoin_ordsucc} : ∀ x0 x1 x2 . nIn x2 x1 ⟶ equip x0 x1 ⟶ equip (ordsucc x0) (binunion x1 (Sing x2))
Definition or^or := λ x0 x1 : ο . ∀ x2 : ο . (x0 ⟶ x2) ⟶ (x1 ⟶ x2) ⟶ x2
Known 1aece.. : ∀ x0 x1 x2 x3 x4 . x4 ∈ SetAdjoin (SetAdjoin (UPair x0 x1) x2) x3 ⟶ ∀ x5 : ι → ο . x5 x0 ⟶ x5 x1 ⟶ x5 x2 ⟶ x5 x3 ⟶ x5 x4
Known orIL^orIL : ∀ x0 x1 : ο . x0 ⟶ or x0 x1
Known orIR^orIR : ∀ x0 x1 : ο . x1 ⟶ or x0 x1
Known d9e1e.. : ∀ x0 x1 x2 x3 . (x0 = x1 ⟶ ∀ x4 : ο . x4) ⟶ (x0 = x2 ⟶ ∀ x4 : ο . x4) ⟶ (x0 = x3 ⟶ ∀ x4 : ο . x4) ⟶ (x1 = x2 ⟶ ∀ x4 : ο . x4) ⟶ (x1 = x3 ⟶ ∀ x4 : ο . x4) ⟶ (x2 = x3 ⟶ ∀ x4 : ο . x4) ⟶ equip (SetAdjoin (SetAdjoin (UPair x0 x1) x2) x3) u4
Theorem 5b07e.. : ∀ x0 x1 x2 x3 x4 . (x0 = x1 ⟶ ∀ x5 : ο . x5) ⟶ (x0 = x2 ⟶ ∀ x5 : ο . x5) ⟶ (x0 = x3 ⟶ ∀ x5 : ο . x5) ⟶ (x0 = x4 ⟶ ∀ x5 : ο . x5) ⟶ (x1 = x2 ⟶ ∀ x5 : ο . x5) ⟶ (x1 = x3 ⟶ ∀ x5 : ο . x5) ⟶ (x1 = x4 ⟶ ∀ x5 : ο . x5) ⟶ (x2 = x3 ⟶ ∀ x5 : ο . x5) ⟶ (x2 = x4 ⟶ ∀ x5 : ο . x5) ⟶ (x3 = x4 ⟶ ∀ x5 : ο . x5) ⟶ equip (SetAdjoin (SetAdjoin (SetAdjoin (UPair x0 x1) x2) x3) x4) u5 (proof)
Param and^and : ο → ο → ο
Param Subq^Subq : ι → ι → ο
Definition u6 := ordsucc u5
Definition u7 := ordsucc u6
Definition u8 := ordsucc u7
Definition u9 := ordsucc u8
Known e041c.. : ∀ x0 : ι → ι → ο . (∀ x1 x2 . x0 x1 x2 ⟶ x0 x2 x1) ⟶ not (or (∀ x1 : ο . (∀ x2 . and (x2 ⊆ u9) (and (equip u3 x2) (∀ x3 . x3 ∈ x2 ⟶ ∀ x4 . x4 ∈ x2 ⟶ (x3 = x4 ⟶ ∀ x5 : ο . x5) ⟶ x0 x3 x4)) ⟶ x1) ⟶ x1) (∀ x1 : ο . (∀ x2 . and (x2 ⊆ u9) (and (equip u4 x2) (∀ x3 . x3 ∈ x2 ⟶ ∀ x4 . x4 ∈ x2 ⟶ (x3 = x4 ⟶ ∀ x5 : ο . x5) ⟶ not (x0 x3 x4))) ⟶ x1) ⟶ x1)) ⟶ ∀ x1 . x1 ∈ u9 ⟶ ∀ x2 : ο . (∀ x3 . x3 ∈ u9 ⟶ ∀ x4 . x4 ∈ u9 ⟶ ∀ x5 . x5 ∈ u9 ⟶ (x1 = x3 ⟶ ∀ x6 : ο . x6) ⟶ (x1 = x4 ⟶ ∀ x6 : ο . x6) ⟶ (x1 = x5 ⟶ ∀ x6 : ο . x6) ⟶ (x3 = x4 ⟶ ∀ x6 : ο . x6) ⟶ (x3 = x5 ⟶ ∀ x6 : ο . x6) ⟶ (x4 = x5 ⟶ ∀ x6 : ο . x6) ⟶ x0 x1 x3 ⟶ x0 x1 x4 ⟶ x0 x1 x5 ⟶ not (x0 x3 x4) ⟶ not (x0 x3 x5) ⟶ not (x0 x4 x5) ⟶ (∀ x6 . x6 ∈ u9 ⟶ x0 x1 x6 ⟶ x6 ∈ SetAdjoin (SetAdjoin (UPair x1 x3) x4) x5) ⟶ x2) ⟶ x2
Known f1644.. : ∀ x0 : ι → ι → ο . (∀ x1 x2 . x0 x1 x2 ⟶ x0 x2 x1) ⟶ not (or (∀ x1 : ο . (∀ x2 . and (x2 ⊆ u9) (and (equip u3 x2) (∀ x3 . x3 ∈ x2 ⟶ ∀ x4 . x4 ∈ x2 ⟶ (x3 = x4 ⟶ ∀ x5 : ο . x5) ⟶ x0 x3 x4)) ⟶ x1) ⟶ x1) (∀ x1 : ο . (∀ x2 . and (x2 ⊆ u9) (and (equip u4 x2) (∀ x3 . x3 ∈ x2 ⟶ ∀ x4 . x4 ∈ x2 ⟶ (x3 = x4 ⟶ ∀ x5 : ο . x5) ⟶ not (x0 x3 x4))) ⟶ x1) ⟶ x1)) ⟶ ∀ x1 . x1 ∈ u9 ⟶ ∀ x2 . x2 ∈ u9 ⟶ (x1 = x2 ⟶ ∀ x3 : ο . x3) ⟶ x0 x1 x2 ⟶ ∀ x3 : ο . (∀ x4 . x4 ∈ u9 ⟶ ∀ x5 . x5 ∈ u9 ⟶ (x1 = x4 ⟶ ∀ x6 : ο . x6) ⟶ (x1 = x5 ⟶ ∀ x6 : ο . x6) ⟶ (x2 = x4 ⟶ ∀ x6 : ο . x6) ⟶ (x2 = x5 ⟶ ∀ x6 : ο . x6) ⟶ (x4 = x5 ⟶ ∀ x6 : ο . x6) ⟶ x0 x1 x4 ⟶ x0 x1 x5 ⟶ not (x0 x2 x4) ⟶ not (x0 x2 x5) ⟶ not (x0 x4 x5) ⟶ (∀ x6 . x6 ∈ u9 ⟶ x0 x1 x6 ⟶ x6 ∈ SetAdjoin (SetAdjoin (UPair x1 x2) x4) x5) ⟶ x3) ⟶ x3
Known xm^xm : ∀ x0 : ο . or x0 (not x0)
Known FalseE^FalseE : False ⟶ ∀ x0 : ο . 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_9^nat_9 : nat_p 9
Definition u10 := ordsucc u9
Known d140d.. : ∀ x0 x1 . x1 ∈ x0 ⟶ ∀ x2 . x2 ∈ x0 ⟶ ∀ x3 . x3 ∈ x0 ⟶ ∀ x4 . x4 ∈ x0 ⟶ ∀ x5 . x5 ∈ x0 ⟶ ∀ x6 . x6 ∈ x0 ⟶ ∀ x7 . x7 ∈ x0 ⟶ ∀ x8 . x8 ∈ x0 ⟶ ∀ x9 . x9 ∈ x0 ⟶ ∀ x10 . x10 ∈ x0 ⟶ (x1 = x2 ⟶ ∀ x11 : ο . x11) ⟶ (x1 = x3 ⟶ ∀ x11 : ο . x11) ⟶ (x2 = x3 ⟶ ∀ x11 : ο . x11) ⟶ (x1 = x4 ⟶ ∀ x11 : ο . x11) ⟶ (x2 = x4 ⟶ ∀ x11 : ο . x11) ⟶ (x3 = x4 ⟶ ∀ x11 : ο . x11) ⟶ (x1 = x5 ⟶ ∀ x11 : ο . x11) ⟶ (x2 = x5 ⟶ ∀ x11 : ο . x11) ⟶ (x3 = x5 ⟶ ∀ x11 : ο . x11) ⟶ (x4 = x5 ⟶ ∀ x11 : ο . x11) ⟶ (x1 = x6 ⟶ ∀ x11 : ο . x11) ⟶ (x2 = x6 ⟶ ∀ x11 : ο . x11) ⟶ (x3 = x6 ⟶ ∀ x11 : ο . x11) ⟶ (x4 = x6 ⟶ ∀ x11 : ο . x11) ⟶ (x5 = x6 ⟶ ∀ x11 : ο . x11) ⟶ (x1 = x7 ⟶ ∀ x11 : ο . x11) ⟶ (x2 = x7 ⟶ ∀ x11 : ο . x11) ⟶ (x3 = x7 ⟶ ∀ x11 : ο . x11) ⟶ (x4 = x7 ⟶ ∀ x11 : ο . x11) ⟶ (x5 = x7 ⟶ ∀ x11 : ο . x11) ⟶ (x6 = x7 ⟶ ∀ x11 : ο . x11) ⟶ (x1 = x8 ⟶ ∀ x11 : ο . x11) ⟶ (x2 = x8 ⟶ ∀ x11 : ο . x11) ⟶ (x3 = x8 ⟶ ∀ x11 : ο . x11) ⟶ (x4 = x8 ⟶ ∀ x11 : ο . x11) ⟶ (x5 = x8 ⟶ ∀ x11 : ο . x11) ⟶ (x6 = x8 ⟶ ∀ x11 : ο . x11) ⟶ (x7 = x8 ⟶ ∀ x11 : ο . x11) ⟶ (x1 = x9 ⟶ ∀ x11 : ο . x11) ⟶ (x2 = x9 ⟶ ∀ x11 : ο . x11) ⟶ (x3 = x9 ⟶ ∀ x11 : ο . x11) ⟶ (x4 = x9 ⟶ ∀ x11 : ο . x11) ⟶ (x5 = x9 ⟶ ∀ x11 : ο . x11) ⟶ (x6 = x9 ⟶ ∀ x11 : ο . x11) ⟶ (x7 = x9 ⟶ ∀ x11 : ο . x11) ⟶ (x8 = x9 ⟶ ∀ x11 : ο . x11) ⟶ (x1 = x10 ⟶ ∀ x11 : ο . x11) ⟶ (x2 = x10 ⟶ ∀ x11 : ο . x11) ⟶ (x3 = x10 ⟶ ∀ x11 : ο . x11) ⟶ (x4 = x10 ⟶ ∀ x11 : ο . x11) ⟶ (x5 = x10 ⟶ ∀ x11 : ο . x11) ⟶ (x6 = x10 ⟶ ∀ x11 : ο . x11) ⟶ (x7 = x10 ⟶ ∀ x11 : ο . x11) ⟶ (x8 = x10 ⟶ ∀ x11 : ο . x11) ⟶ (x9 = x10 ⟶ ∀ x11 : ο . x11) ⟶ atleastp u10 x0
Known neq_i_sym^neq_i_sym : ∀ x0 x1 . (x0 = x1 ⟶ ∀ x2 : ο . x2) ⟶ x1 = x0 ⟶ ∀ x2 : ο . x2
Known binunionE^binunionE : ∀ x0 x1 x2 . x2 ∈ binunion x0 x1 ⟶ or (x2 ∈ x0) (x2 ∈ x1)
Known UPairE^UPairE : ∀ x0 x1 x2 . x0 ∈ UPair x1 x2 ⟶ or (x0 = x1) (x0 = x2)
Known binunionI1^binunionI1 : ∀ x0 x1 x2 . x2 ∈ x0 ⟶ x2 ∈ binunion x0 x1
Known UPairI1^UPairI1 : ∀ x0 x1 . x0 ∈ UPair x0 x1
Known binunionI2^binunionI2 : ∀ x0 x1 x2 . x2 ∈ x1 ⟶ x2 ∈ binunion x0 x1
Known SingI^SingI : ∀ x0 . x0 ∈ Sing x0
Known SingE^SingE : ∀ x0 x1 . x1 ∈ Sing x0 ⟶ x1 = x0
Known UPairI2^UPairI2 : ∀ x0 x1 . x1 ∈ UPair x0 x1
Theorem bd9af.. : ∀ x0 : ι → ι → ο . (∀ x1 x2 . x0 x1 x2 ⟶ x0 x2 x1) ⟶ not (or (∀ x1 : ο . (∀ x2 . and (x2 ⊆ u9) (and (equip u3 x2) (∀ x3 . x3 ∈ x2 ⟶ ∀ x4 . x4 ∈ x2 ⟶ (x3 = x4 ⟶ ∀ x5 : ο . x5) ⟶ x0 x3 x4)) ⟶ x1) ⟶ x1) (∀ x1 : ο . (∀ x2 . and (x2 ⊆ u9) (and (equip u4 x2) (∀ x3 . x3 ∈ x2 ⟶ ∀ x4 . x4 ∈ x2 ⟶ (x3 = x4 ⟶ ∀ x5 : ο . x5) ⟶ not (x0 x3 x4))) ⟶ x1) ⟶ x1)) ⟶ ∀ x1 . x1 ∈ u9 ⟶ ∀ x2 : ο . (∀ x3 . x3 ∈ u9 ⟶ ∀ x4 . x4 ∈ u9 ⟶ ∀ x5 . x5 ∈ u9 ⟶ ∀ x6 . x6 ∈ u9 ⟶ (x1 = x3 ⟶ ∀ x7 : ο . x7) ⟶ (x1 = x4 ⟶ ∀ x7 : ο . x7) ⟶ (x1 = x5 ⟶ ∀ x7 : ο . x7) ⟶ (x1 = x6 ⟶ ∀ x7 : ο . x7) ⟶ (x3 = x4 ⟶ ∀ x7 : ο . x7) ⟶ (x3 = x5 ⟶ ∀ x7 : ο . x7) ⟶ (x3 = x6 ⟶ ∀ x7 : ο . x7) ⟶ (x4 = x5 ⟶ ∀ x7 : ο . x7) ⟶ (x4 = x6 ⟶ ∀ x7 : ο . x7) ⟶ (x5 = x6 ⟶ ∀ x7 : ο . x7) ⟶ x0 x1 x3 ⟶ x0 x1 x4 ⟶ x0 x1 x5 ⟶ not (x0 x3 x4) ⟶ not (x0 x3 x5) ⟶ not (x0 x4 x5) ⟶ (∀ x7 . x7 ∈ u9 ⟶ x0 x1 x7 ⟶ x7 ∈ SetAdjoin (SetAdjoin (UPair x1 x3) x4) x5) ⟶ x0 x6 x3 ⟶ x0 x6 x4 ⟶ x2) ⟶ x2 (proof)