Proofgold Address

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
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 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 SingE^SingE : ∀ x0 x1 . x1 ∈ Sing x0 ⟶ x1 = x0
Theorem 3cea6.. : ∀ x0 x1 x2 x3 x4 x5 . x5 ∈ SetAdjoin (SetAdjoin (SetAdjoin (UPair x0 x1) x2) x3) x4 ⟶ ∀ x6 : ι → ο . x6 x0 ⟶ x6 x1 ⟶ x6 x2 ⟶ x6 x3 ⟶ x6 x4 ⟶ x6 x5 (proof)
Definition False^False := ∀ x0 : ο . x0
Definition not^not := λ x0 : ο . x0 ⟶ False
Definition and^and := λ x0 x1 : ο . ∀ x2 : ο . (x0 ⟶ x1 ⟶ x2) ⟶ x2
Definition Subq^Subq := λ x0 x1 . ∀ x2 . x2 ∈ x0 ⟶ x2 ∈ x1
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 u6 := ordsucc u5
Definition u7 := ordsucc u6
Definition u8 := ordsucc u7
Definition u9 := ordsucc u8
Param equip^equip : ι → ι → ο
Param setminus^setminus : ι → ι → ι
Param nat_p^nat_p : ι → ο
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
Known 48e0f.. : ∀ x0 . nat_p x0 ⟶ ∀ x1 . or (atleastp x1 x0) (atleastp (ordsucc x0) x1)
Known nat_3^nat_3 : nat_p 3
Known 4fb58..^{Pigeonhole_not_atleastp_ordsucc} : ∀ x0 . nat_p x0 ⟶ not (atleastp (ordsucc x0) x0)
Known nat_8^nat_8 : nat_p 8
Param binintersect^binintersect : ι → ι → ι
Known a8a92.. : ∀ x0 x1 . x0 = binunion (setminus x0 x1) (binintersect x0 x1)
Known binintersect_com^{binintersect_com} : ∀ x0 x1 . binintersect x0 x1 = binintersect x1 x0
Known binintersect_Subq_eq_1^{binintersect_Subq_eq_1} : ∀ x0 x1 . x0 ⊆ x1 ⟶ binintersect x0 x1 = x0
Param setsum^setsum : ι → ι → ι
Known atleastp_tra^atleastp_tra : ∀ x0 x1 x2 . atleastp x0 x1 ⟶ atleastp x1 x2 ⟶ atleastp x0 x2
Definition nIn^nIn := λ x0 x1 . not (x0 ∈ x1)
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 equip_atleastp^{equip_atleastp} : ∀ x0 x1 . equip x0 x1 ⟶ atleastp x0 x1
Known 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
Known setminusE2^setminusE2 : ∀ x0 x1 x2 . x2 ∈ setminus x0 x1 ⟶ nIn x2 x1
Known equip_sym^equip_sym : ∀ x0 x1 . equip x0 x1 ⟶ equip x1 x0
Param add_nat^add_nat : ι → ι → ι
Known e705e.. : add_nat u5 u3 = u8
Known add_nat_com^add_nat_com : ∀ x0 . nat_p x0 ⟶ ∀ x1 . nat_p x1 ⟶ add_nat x0 x1 = add_nat x1 x0
Known nat_5^nat_5 : nat_p 5
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 equip_ref^equip_ref : ∀ x0 . equip x0 x0
Param If_i^If_i : ο → ι → ι → ι
Known andI^andI : ∀ x0 x1 : ο . x0 ⟶ x1 ⟶ and x0 x1
Known If_i_or^If_i_or : ∀ x0 : ο . ∀ x1 x2 . or (If_i x0 x1 x2 = x1) (If_i x0 x1 x2 = x2)
Known In_0_3^In_0_3 : 0 ∈ 3
Known In_1_3^In_1_3 : 1 ∈ 3
Known In_2_3^In_2_3 : 2 ∈ 3
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 If_i_1^If_i_1 : ∀ x0 : ο . ∀ x1 x2 . x0 ⟶ If_i x0 x1 x2 = x1
Known 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
Known FalseE^FalseE : False ⟶ ∀ x0 : ο . x0
Known If_i_0^If_i_0 : ∀ x0 : ο . ∀ x1 x2 . not x0 ⟶ If_i x0 x1 x2 = x2
Known neq_0_1^neq_0_1 : 0 = 1 ⟶ ∀ x0 : ο . x0
Known neq_0_2^neq_0_2 : 0 = 2 ⟶ ∀ x0 : ο . x0
Known neq_1_0^neq_1_0 : u1 = 0 ⟶ ∀ x0 : ο . x0
Known neq_2_0^neq_2_0 : u2 = 0 ⟶ ∀ x0 : ο . x0
Known neq_1_2^neq_1_2 : 1 = 2 ⟶ ∀ x0 : ο . x0
Known neq_2_1^neq_2_1 : u2 = u1 ⟶ ∀ x0 : ο . x0
Known binunionI2^binunionI2 : ∀ x0 x1 x2 . x2 ∈ x1 ⟶ x2 ∈ binunion x0 x1
Known SingI^SingI : ∀ x0 . x0 ∈ Sing x0
Known binunionI1^binunionI1 : ∀ x0 x1 x2 . x2 ∈ x0 ⟶ x2 ∈ binunion x0 x1
Known UPairI2^UPairI2 : ∀ x0 x1 . x1 ∈ UPair x0 x1
Known UPairI1^UPairI1 : ∀ x0 x1 . x0 ∈ UPair x0 x1
Known neq_i_sym^neq_i_sym : ∀ x0 x1 . (x0 = x1 ⟶ ∀ x2 : ο . x2) ⟶ x1 = x0 ⟶ ∀ x2 : ο . x2
Known 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
Known orIR^orIR : ∀ x0 x1 : ο . x1 ⟶ or x0 x1
Theorem 0799b.. : ∀ 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 ⟶ ∀ x5 . x5 ∈ u9 ⟶ (x1 = x2 ⟶ ∀ x6 : ο . x6) ⟶ (x1 = x3 ⟶ ∀ x6 : ο . x6) ⟶ (x1 = x4 ⟶ ∀ x6 : ο . x6) ⟶ (x1 = x5 ⟶ ∀ x6 : ο . x6) ⟶ (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) ⟶ x0 x1 x2 ⟶ x0 x1 x3 ⟶ x0 x1 x4 ⟶ not (x0 x2 x3) ⟶ not (x0 x2 x4) ⟶ not (x0 x3 x4) ⟶ x0 x5 x2 ⟶ x0 x5 x3 ⟶ x0 x5 x4 ⟶ False (proof)