Proofgold Asset

Param bij^bij : ι → ι → (ι → ι) → ο
Definition equip^equip := λ x0 x1 . ∀ x2 : ο . (∀ x3 : ι → ι . bij x0 x1 x3 ⟶ x2) ⟶ x2
Definition False^False := ∀ x0 : ο . x0
Definition not^not := λ x0 : ο . x0 ⟶ False
Definition nIn^nIn := λ x0 x1 . not (x0 ∈ x1)
Param binunion^binunion : ι → ι → ι
Param setsum^setsum : ι → ι → ι
Definition and^and := λ x0 x1 : ο . ∀ x2 : ο . (x0 ⟶ x1 ⟶ x2) ⟶ x2
Known bijE^bijE : ∀ x0 x1 . ∀ x2 : ι → ι . bij x0 x1 x2 ⟶ ∀ x3 : ο . ((∀ x4 . x4 ∈ x0 ⟶ x2 x4 ∈ x1) ⟶ (∀ x4 . x4 ∈ x0 ⟶ ∀ x5 . x5 ∈ x0 ⟶ x2 x4 = x2 x5 ⟶ x4 = x5) ⟶ (∀ x4 . x4 ∈ x1 ⟶ ∀ x5 : ο . (∀ x6 . and (x6 ∈ x0) (x2 x6 = x4) ⟶ x5) ⟶ x5) ⟶ x3) ⟶ x3
Param If_i^If_i : ο → ι → ι → ι
Param Inj0^Inj0 : ι → ι
Param Inj1^Inj1 : ι → ι
Known bijI^bijI : ∀ x0 x1 . ∀ x2 : ι → ι . (∀ x3 . x3 ∈ x0 ⟶ x2 x3 ∈ x1) ⟶ (∀ x3 . x3 ∈ x0 ⟶ ∀ x4 . x4 ∈ x0 ⟶ x2 x3 = x2 x4 ⟶ x3 = x4) ⟶ (∀ x3 . x3 ∈ x1 ⟶ ∀ x4 : ο . (∀ x5 . and (x5 ∈ x0) (x2 x5 = x3) ⟶ x4) ⟶ x4) ⟶ bij x0 x1 x2
Definition or^or := λ x0 x1 : ο . ∀ x2 : ο . (x0 ⟶ x2) ⟶ (x1 ⟶ x2) ⟶ x2
Known binunionE^binunionE : ∀ x0 x1 x2 . x2 ∈ binunion x0 x1 ⟶ or (x2 ∈ x0) (x2 ∈ x1)
Known If_i_1^If_i_1 : ∀ x0 : ο . ∀ x1 x2 . x0 ⟶ If_i x0 x1 x2 = x1
Known Inj0_setsum^Inj0_setsum : ∀ x0 x1 x2 . x2 ∈ x0 ⟶ Inj0 x2 ∈ setsum x0 x1
Known If_i_0^If_i_0 : ∀ x0 : ο . ∀ x1 x2 . not x0 ⟶ If_i x0 x1 x2 = x2
Known Inj1_setsum^Inj1_setsum : ∀ x0 x1 x2 . x2 ∈ x1 ⟶ Inj1 x2 ∈ setsum x0 x1
Known Inj0_inj^Inj0_inj : ∀ x0 x1 . Inj0 x0 = Inj0 x1 ⟶ x0 = x1
Known FalseE^FalseE : False ⟶ ∀ x0 : ο . x0
Known Inj0_Inj1_neq^{Inj0_Inj1_neq} : ∀ x0 x1 . Inj0 x0 = Inj1 x1 ⟶ ∀ x2 : ο . x2
Known Inj1_inj^Inj1_inj : ∀ x0 x1 . Inj1 x0 = Inj1 x1 ⟶ x0 = x1
Known f4c7c.. : ∀ x0 x1 . ∀ x2 : ι → ο . (∀ x3 . x3 ∈ x0 ⟶ x2 (Inj0 x3)) ⟶ (∀ x3 . x3 ∈ x1 ⟶ x2 (Inj1 x3)) ⟶ ∀ x3 . x3 ∈ setsum x0 x1 ⟶ x2 x3
Known andI^andI : ∀ x0 x1 : ο . x0 ⟶ x1 ⟶ and x0 x1
Known binunionI1^binunionI1 : ∀ x0 x1 x2 . x2 ∈ x0 ⟶ x2 ∈ binunion x0 x1
Known binunionI2^binunionI2 : ∀ x0 x1 x2 . x2 ∈ x1 ⟶ x2 ∈ binunion x0 x1
Theorem d778e.. : ∀ x0 x1 x2 x3 . equip x0 x2 ⟶ equip x1 x3 ⟶ (∀ x4 . x4 ∈ x0 ⟶ nIn x4 x1) ⟶ equip (binunion x0 x1) (setsum x2 x3) (proof)
Param add_nat^add_nat : ι → ι → ι
Param ordsucc^ordsucc : ι → ι
Definition u1 := 1
Definition u2 := ordsucc u1
Definition u3 := ordsucc u2
Definition u4 := ordsucc u3
Param nat_p^nat_p : ι → ο
Known add_nat_SR^add_nat_SR : ∀ x0 x1 . nat_p x1 ⟶ add_nat x0 (ordsucc x1) = ordsucc (add_nat x0 x1)
Known nat_0^nat_0 : nat_p 0
Known add_nat_0R^add_nat_0R : ∀ x0 . add_nat x0 0 = x0
Theorem 480b2.. : add_nat u3 u1 = u4 (proof)
Definition u5 := ordsucc u4
Definition u6 := ordsucc u5
Theorem 561b1.. : add_nat u5 u1 = u6 (proof)
Definition u7 := ordsucc u6
Known nat_1^nat_1 : nat_p 1
Theorem bd216.. : add_nat u5 u2 = u7 (proof)
Definition u8 := ordsucc u7
Known nat_2^nat_2 : nat_p 2
Theorem e705e.. : add_nat u5 u3 = u8 (proof)
Definition Subq^Subq := λ x0 x1 . ∀ x2 . x2 ∈ x0 ⟶ x2 ∈ x1
Param setminus^setminus : ι → ι → ι
Param binintersect^binintersect : ι → ι → ι
Known Subq_binintersection_eq^{Subq_binintersection_eq} : ∀ x0 x1 . x0 ⊆ x1 = (binintersect x0 x1 = x0)
Known a8a92.. : ∀ x0 x1 . x0 = binunion (setminus x0 x1) (binintersect x0 x1)
Known binunion_com^binunion_com : ∀ x0 x1 . binunion x0 x1 = binunion x1 x0
Known binintersect_com^{binintersect_com} : ∀ x0 x1 . binintersect x0 x1 = binintersect x1 x0
Theorem 80238.. : ∀ x0 x1 . x0 ⊆ x1 ⟶ x1 = binunion x0 (setminus x1 x0) (proof)
Param UPair^UPair : ι → ι → ι
Param Sing^Sing : ι → ι
Known set_ext^set_ext : ∀ x0 x1 . x0 ⊆ x1 ⟶ x1 ⊆ x0 ⟶ x0 = x1
Known UPairE^UPairE : ∀ x0 x1 x2 . x0 ∈ UPair x1 x2 ⟶ or (x0 = x1) (x0 = x2)
Known SingI^SingI : ∀ x0 . x0 ∈ Sing x0
Known SingE^SingE : ∀ x0 x1 . x1 ∈ Sing x0 ⟶ x1 = x0
Known UPairI1^UPairI1 : ∀ x0 x1 . x0 ∈ UPair x0 x1
Known UPairI2^UPairI2 : ∀ x0 x1 . x1 ∈ UPair x0 x1
Theorem cbaf1.. : ∀ x0 x1 . UPair x0 x1 = binunion (Sing x0) (Sing x1) (proof)
Known equip_sym^equip_sym : ∀ x0 x1 . equip x0 x1 ⟶ equip x1 x0
Known e8bc0..^{equip_adjoin_ordsucc} : ∀ x0 x1 x2 . nIn x2 x1 ⟶ equip x0 x1 ⟶ equip (ordsucc x0) (binunion x1 (Sing x2))
Known 5169f..^equip_Sing_1 : ∀ x0 . equip (Sing x0) u1
Theorem ced33.. : ∀ x0 x1 . (x0 = x1 ⟶ ∀ x2 : ο . x2) ⟶ equip (UPair x0 x1) u2 (proof)
Definition SetAdjoin^SetAdjoin := λ x0 x1 . binunion x0 (Sing x1)
Theorem 76c0f.. : ∀ x0 x1 x2 x3 . x3 ∈ SetAdjoin (UPair x0 x1) x2 ⟶ ∀ x4 : ι → ο . x4 x0 ⟶ x4 x1 ⟶ x4 x2 ⟶ x4 x3 (proof)
Theorem 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 (proof)
Theorem a515c.. : ∀ x0 x1 x2 . (x0 = x1 ⟶ ∀ x3 : ο . x3) ⟶ (x0 = x2 ⟶ ∀ x3 : ο . x3) ⟶ (x1 = x2 ⟶ ∀ x3 : ο . x3) ⟶ equip (SetAdjoin (UPair x0 x1) x2) u3 (proof)
Known orIL^orIL : ∀ x0 x1 : ο . x0 ⟶ or x0 x1
Known orIR^orIR : ∀ x0 x1 : ο . x1 ⟶ or x0 x1
Theorem 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 (proof)
Definition u9 := ordsucc u8
Known dneg^dneg : ∀ x0 : ο . not (not x0) ⟶ x0
Theorem bf767.. : ∀ x0 : ι → ι → ο . (∀ x1 x2 . x0 x1 x2 ⟶ x0 x2 x1) ⟶ not (∀ 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 . x1 ∈ u9 ⟶ ∀ x2 . x2 ∈ u9 ⟶ ∀ x3 . x3 ∈ u9 ⟶ x0 x1 x2 ⟶ x0 x1 x3 ⟶ x0 x2 x3 ⟶ ∀ x4 : ο . (x1 = x2 ⟶ x4) ⟶ (x1 = x3 ⟶ x4) ⟶ (x2 = x3 ⟶ x4) ⟶ x4 (proof)
Theorem 0728d.. : ∀ x0 : ι → ι → ο . (∀ x1 x2 . x0 x1 x2 ⟶ x0 x2 x1) ⟶ not (∀ 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 ⟶ ∀ x3 . x3 ∈ u9 ⟶ ∀ x4 . x4 ∈ u9 ⟶ not (x0 x1 x2) ⟶ not (x0 x1 x3) ⟶ not (x0 x1 x4) ⟶ not (x0 x2 x3) ⟶ not (x0 x2 x4) ⟶ not (x0 x3 x4) ⟶ ∀ x5 : ο . (x1 = x2 ⟶ x5) ⟶ (x1 = x3 ⟶ x5) ⟶ (x1 = x4 ⟶ x5) ⟶ (x2 = x3 ⟶ x5) ⟶ (x2 = x4 ⟶ x5) ⟶ (x3 = x4 ⟶ x5) ⟶ x5 (proof)
Param atleastp^atleastp : ι → ι → ο
Definition TwoRamseyProp^{TwoRamseyProp} := λ x0 x1 x2 . ∀ x3 : ι → ι → ο . (∀ x4 x5 . x3 x4 x5 ⟶ x3 x5 x4) ⟶ or (∀ x4 : ο . (∀ x5 . and (x5 ⊆ x2) (and (equip x0 x5) (∀ x6 . x6 ∈ x5 ⟶ ∀ x7 . x7 ∈ x5 ⟶ (x6 = x7 ⟶ ∀ x8 : ο . x8) ⟶ x3 x6 x7)) ⟶ x4) ⟶ x4) (∀ x4 : ο . (∀ x5 . and (x5 ⊆ x2) (and (equip x1 x5) (∀ x6 . x6 ∈ x5 ⟶ ∀ x7 . x7 ∈ x5 ⟶ (x6 = x7 ⟶ ∀ x8 : ο . x8) ⟶ not (x3 x6 x7))) ⟶ x4) ⟶ x4)
Known Subq_tra^Subq_tra : ∀ x0 x1 x2 . x0 ⊆ x1 ⟶ x1 ⊆ x2 ⟶ x0 ⊆ x2
Known binunion_Subq_min^{binunion_Subq_min} : ∀ x0 x1 x2 . x0 ⊆ x2 ⟶ x1 ⊆ x2 ⟶ binunion x0 x1 ⊆ x2
Known equip_tra^equip_tra : ∀ x0 x1 x2 . equip x0 x1 ⟶ equip x1 x2 ⟶ equip x0 x2
Known c88e0.. : ∀ x0 x1 x2 x3 . nat_p x0 ⟶ nat_p x1 ⟶ equip x0 x2 ⟶ equip x1 x3 ⟶ equip (add_nat x0 x1) (setsum x2 x3)
Known nat_3^nat_3 : nat_p 3
Known equip_ref^equip_ref : ∀ x0 . equip x0 x0
Known 46dcf.. : ∀ x0 x1 x2 x3 . atleastp x2 x3 ⟶ TwoRamseyProp x0 x1 x2 ⟶ TwoRamseyProp x0 x1 x3
Known TwoRamseyProp_3_3_6^{TwoRamseyProp_3_3_6} : TwoRamseyProp 3 3 6
Param Sep^Sep : ι → (ι → ο) → ι
Known and3I^and3I : ∀ x0 x1 x2 : ο . x0 ⟶ x1 ⟶ x2 ⟶ and (and x0 x1) x2
Known 48e0f.. : ∀ x0 . nat_p x0 ⟶ ∀ x1 . or (atleastp x1 x0) (atleastp (ordsucc x0) x1)
Known nat_5^nat_5 : nat_p 5
Known 4fb58..^{Pigeonhole_not_atleastp_ordsucc} : ∀ x0 . nat_p x0 ⟶ not (atleastp (ordsucc x0) x0)
Known nat_8^nat_8 : nat_p 8
Known atleastp_tra^atleastp_tra : ∀ x0 x1 x2 . atleastp x0 x1 ⟶ atleastp x1 x2 ⟶ atleastp x0 x2
Known 385ef.. : ∀ x0 x1 x2 x3 . atleastp x0 x2 ⟶ atleastp x1 x3 ⟶ (∀ x4 . x4 ∈ x0 ⟶ nIn x4 x1) ⟶ atleastp (binunion x0 x1) (setsum x2 x3)
Known setminusE2^setminusE2 : ∀ x0 x1 x2 . x2 ∈ setminus x0 x1 ⟶ nIn x2 x1
Known equip_atleastp^{equip_atleastp} : ∀ x0 x1 . equip x0 x1 ⟶ atleastp x0 x1
Known d03c6.. : ∀ x0 . atleastp u4 x0 ⟶ ∀ x1 : ο . (∀ x2 . x2 ∈ x0 ⟶ ∀ x3 . x3 ∈ x0 ⟶ ∀ x4 . x4 ∈ x0 ⟶ ∀ x5 . x5 ∈ x0 ⟶ (x2 = x3 ⟶ ∀ x6 : ο . x6) ⟶ (x2 = x4 ⟶ ∀ x6 : ο . x6) ⟶ (x2 = x5 ⟶ ∀ x6 : ο . x6) ⟶ (x3 = x4 ⟶ ∀ x6 : ο . x6) ⟶ (x3 = x5 ⟶ ∀ x6 : ο . x6) ⟶ (x4 = x5 ⟶ ∀ x6 : ο . x6) ⟶ x1) ⟶ x1
Known setminusE^setminusE : ∀ x0 x1 x2 . x2 ∈ setminus x0 x1 ⟶ and (x2 ∈ x0) (nIn x2 x1)
Known xm^xm : ∀ x0 : ο . or x0 (not x0)
Known SepI^SepI : ∀ x0 . ∀ x1 : ι → ο . ∀ x2 . x2 ∈ x0 ⟶ x1 x2 ⟶ x2 ∈ Sep x0 x1
Known SepE2^SepE2 : ∀ x0 . ∀ x1 : ι → ο . ∀ x2 . x2 ∈ Sep x0 x1 ⟶ x1 x2
Known Sep_Subq^Sep_Subq : ∀ x0 . ∀ x1 : ι → ο . Sep x0 x1 ⊆ x0
Theorem e5c2b.. : ∀ 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 ⟶ x2) ⟶ x2 (proof)
Theorem 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 (proof)
Known neq_i_sym^neq_i_sym : ∀ x0 x1 . (x0 = x1 ⟶ ∀ x2 : ο . x2) ⟶ x1 = x0 ⟶ ∀ x2 : ο . x2
Theorem 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 (proof)
Theorem 8455a.. : ∀ 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 ⟶ ∀ x3 . x3 ∈ u9 ⟶ (x1 = x2 ⟶ ∀ x4 : ο . x4) ⟶ (x1 = x3 ⟶ ∀ x4 : ο . x4) ⟶ (x2 = x3 ⟶ ∀ x4 : ο . x4) ⟶ x0 x1 x2 ⟶ x0 x1 x3 ⟶ ∀ x4 : ο . (∀ x5 . x5 ∈ u9 ⟶ (x1 = x5 ⟶ ∀ x6 : ο . x6) ⟶ (x2 = x5 ⟶ ∀ x6 : ο . x6) ⟶ (x3 = x5 ⟶ ∀ x6 : ο . x6) ⟶ x0 x1 x5 ⟶ not (x0 x2 x5) ⟶ not (x0 x3 x5) ⟶ (∀ x6 . x6 ∈ u9 ⟶ x0 x1 x6 ⟶ x6 ∈ SetAdjoin (SetAdjoin (UPair x1 x2) x3) x5) ⟶ x4) ⟶ x4 (proof)
Theorem c62d8.. : ∀ 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 ⟶ ∀ x3 . x3 ∈ u9 ⟶ ∀ x4 . x4 ∈ u9 ⟶ (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) ⟶ x0 x1 x2 ⟶ x0 x1 x3 ⟶ x0 x1 x4 ⟶ ∀ x5 . x5 ∈ u9 ⟶ x0 x1 x5 ⟶ x5 ∈ SetAdjoin (SetAdjoin (UPair x1 x2) x3) x4 (proof)