Proofgold Address

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
Definition u10 := ordsucc u9
Definition u11 := ordsucc u10
Definition u12 := ordsucc u11
Definition u13 := ordsucc u12
Definition u14 := ordsucc u13
Definition u15 := ordsucc u14
Definition Subq^Subq := λ x0 x1 . ∀ x2 . x2 ∈ x0 ⟶ x2 ∈ x1
Definition u16 := ordsucc u15
Param atleastp^atleastp : ι → ι → ο
Definition False^False := ∀ x0 : ο . x0
Definition not^not := λ x0 : ο . x0 ⟶ False
Known 1c7b4.. : ∀ x0 : ι → ι → ο . x0 0 u2 ⟶ x0 u4 u6 ⟶ x0 u1 u12 ⟶ x0 u5 u12 ⟶ x0 u8 u12 ⟶ x0 u9 u12 ⟶ x0 u3 u13 ⟶ x0 u7 u13 ⟶ x0 u10 u13 ⟶ x0 u2 u14 ⟶ x0 u6 u14 ⟶ x0 u11 u14 ⟶ x0 0 u15 ⟶ x0 u4 u15 ⟶ ∀ x1 . x1 ⊆ u16 ⟶ atleastp u6 x1 ⟶ u12 ∈ x1 ⟶ u13 ∈ x1 ⟶ u14 ∈ x1 ⟶ (∀ x2 . x2 ∈ x1 ⟶ ∀ x3 . x3 ∈ x1 ⟶ not (x0 x2 x3)) ⟶ False
Param binintersect^binintersect : ι → ι → ι
Definition nIn^nIn := λ x0 x1 . not (x0 ∈ x1)
Definition and^and := λ x0 x1 : ο . ∀ x2 : ο . (x0 ⟶ x1 ⟶ x2) ⟶ x2
Param add_nat^add_nat : ι → ι → ι
Definition or^or := λ x0 x1 : ο . ∀ x2 : ο . (x0 ⟶ x2) ⟶ (x1 ⟶ x2) ⟶ x2
Known xm^xm : ∀ x0 : ο . or x0 (not x0)
Known f03aa.. : ∀ x0 . atleastp 3 x0 ⟶ ∀ x1 : ο . (∀ x2 . x2 ∈ x0 ⟶ ∀ x3 . x3 ∈ x0 ⟶ ∀ x4 . x4 ∈ x0 ⟶ (x2 = x3 ⟶ ∀ x5 : ο . x5) ⟶ (x2 = x4 ⟶ ∀ x5 : ο . x5) ⟶ (x3 = x4 ⟶ ∀ x5 : ο . x5) ⟶ x1) ⟶ x1
Known binintersectE1^{binintersectE1} : ∀ x0 x1 x2 . x2 ∈ binintersect x0 x1 ⟶ x2 ∈ x0
Known FalseE^FalseE : False ⟶ ∀ x0 : ο . x0
Param nat_p^nat_p : ι → ο
Known 48e0f.. : ∀ x0 . nat_p x0 ⟶ ∀ x1 . or (atleastp x1 x0) (atleastp (ordsucc x0) x1)
Known nat_2^nat_2 : nat_p 2
Known 4fb58..^{Pigeonhole_not_atleastp_ordsucc} : ∀ x0 . nat_p x0 ⟶ not (atleastp (ordsucc x0) x0)
Known nat_5^nat_5 : nat_p 5
Known atleastp_tra^atleastp_tra : ∀ x0 x1 x2 . atleastp x0 x1 ⟶ atleastp x1 x2 ⟶ atleastp x0 x2
Param binunion^binunion : ι → ι → ι
Param Sing^Sing : ι → ι
Param setsum^setsum : ι → ι → ι
Known nat_setsum_ordsucc^{nat_setsum1_ordsucc} : ∀ x0 . nat_p x0 ⟶ setsum 1 x0 = ordsucc x0
Known nat_4^nat_4 : nat_p 4
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 4f2c3.. : ∀ x0 . atleastp (Sing x0) u1
Known nat_3^nat_3 : nat_p 3
Known In_no4cycle^In_no4cycle : ∀ x0 x1 x2 x3 . x0 ∈ x1 ⟶ x1 ∈ x2 ⟶ x2 ∈ x3 ⟶ x3 ∈ x0 ⟶ False
Known f6a92.. : 12 ∈ 13
Known 3ef99.. : 13 ∈ 14
Known 2e90c.. : 14 ∈ 15
Known SingE^SingE : ∀ x0 x1 . x1 ∈ Sing x0 ⟶ x1 = x0
Known binintersectE2^{binintersectE2} : ∀ x0 x1 x2 . x2 ∈ binintersect x0 x1 ⟶ x2 ∈ x1
Known binunionE^binunionE : ∀ x0 x1 x2 . x2 ∈ binunion x0 x1 ⟶ or (x2 ∈ x0) (x2 ∈ x1)
Known b8e82.. : u15 = u14 ⟶ ∀ x0 : ο . x0
Known In_no3cycle^In_no3cycle : ∀ x0 x1 x2 . x0 ∈ x1 ⟶ x1 ∈ x2 ⟶ x2 ∈ x0 ⟶ False
Known ef4da.. : u14 = u12 ⟶ ∀ x0 : ο . x0
Known 72647.. : u15 = u12 ⟶ ∀ x0 : ο . x0
Known In_irref^In_irref : ∀ x0 . nIn x0 x0
Known set_ext^set_ext : ∀ x0 x1 . x0 ⊆ x1 ⟶ x1 ⊆ x0 ⟶ x0 = x1
Known binunionI2^binunionI2 : ∀ x0 x1 x2 . x2 ∈ x1 ⟶ x2 ∈ binunion x0 x1
Known binintersectI^{binintersectI} : ∀ x0 x1 x2 . x2 ∈ x0 ⟶ x2 ∈ x1 ⟶ x2 ∈ binintersect x0 x1
Known binunionI1^binunionI1 : ∀ x0 x1 x2 . x2 ∈ x0 ⟶ x2 ∈ binunion x0 x1
Known SingI^SingI : ∀ x0 . x0 ∈ Sing x0
Known nat_In_atleastp : ∀ x0 . nat_p x0 ⟶ ∀ x1 . x1 ∈ x0 ⟶ atleastp x1 x0
Known In_2_3^In_2_3 : 2 ∈ 3
Known Eps_i_ex^Eps_i_ex : ∀ x0 : ι → ο . (∀ x1 : ο . (∀ x2 . x0 x2 ⟶ x1) ⟶ x1) ⟶ x0 (prim0 x0)
Known andI^andI : ∀ x0 x1 : ο . x0 ⟶ x1 ⟶ and x0 x1
Known In_0_5^In_0_5 : 0 ∈ 5
Known add_nat_0L^add_nat_0L : ∀ x0 . nat_p x0 ⟶ add_nat 0 x0 = x0
Known nat_12^nat_12 : nat_p 12
Known In_1_5^In_1_5 : 1 ∈ 5
Known add_nat_SL^add_nat_SL : ∀ x0 . nat_p x0 ⟶ ∀ x1 . nat_p x1 ⟶ add_nat (ordsucc x0) x1 = ordsucc (add_nat x0 x1)
Known nat_0^nat_0 : nat_p 0
Known In_2_5^In_2_5 : 2 ∈ 5
Known nat_1^nat_1 : nat_p 1
Known In_3_5^In_3_5 : 3 ∈ 5
Known binintersectE^{binintersectE} : ∀ x0 x1 x2 . x2 ∈ binintersect x0 x1 ⟶ and (x2 ∈ x0) (x2 ∈ x1)
Known ordsuccE^ordsuccE : ∀ x0 x1 . x1 ∈ ordsucc x0 ⟶ or (x1 ∈ x0) (x1 = x0)
Known cases_9^cases_9 : ∀ x0 . x0 ∈ 9 ⟶ ∀ x1 : ι → ο . x1 0 ⟶ x1 1 ⟶ x1 2 ⟶ x1 3 ⟶ x1 4 ⟶ x1 5 ⟶ x1 6 ⟶ x1 7 ⟶ x1 8 ⟶ x1 x0
Theorem 6b8c1.. : ∀ x0 : ι → ι → ο . x0 0 u2 ⟶ x0 u4 u6 ⟶ x0 u1 u12 ⟶ x0 u5 u12 ⟶ x0 u8 u12 ⟶ x0 u9 u12 ⟶ x0 u3 u13 ⟶ x0 u7 u13 ⟶ x0 u10 u13 ⟶ x0 u2 u14 ⟶ x0 u6 u14 ⟶ x0 u11 u14 ⟶ x0 0 u15 ⟶ x0 u4 u15 ⟶ x0 u3 u4 ⟶ x0 0 u7 ⟶ x0 u10 u14 ⟶ ∀ x1 . x1 ⊆ u16 ⟶ atleastp u6 x1 ⟶ u12 ∈ x1 ⟶ u14 ∈ x1 ⟶ (∀ x2 . x2 ∈ x1 ⟶ ∀ x3 . x3 ∈ x1 ⟶ not (x0 x2 x3)) ⟶ False (proof)