Proofgold Address

Definition and^and := λ x0 x1 : ο . ∀ x2 : ο . (x0 ⟶ x1 ⟶ x2) ⟶ x2
Param bij^bij : ι → ι → (ι → ι) → ο
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
Param equip^equip : ι → ι → ο
Known equip_ref^equip_ref : ∀ x0 . equip x0 x0
Definition or^or := λ x0 x1 : ο . ∀ x2 : ο . (x0 ⟶ x2) ⟶ (x1 ⟶ x2) ⟶ x2
Definition Subq^Subq := λ x0 x1 . ∀ x2 . x2 ∈ x0 ⟶ x2 ∈ x1
Definition False^False := ∀ x0 : ο . x0
Definition not^not := λ x0 : ο . x0 ⟶ False
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)
Param nat_p^nat_p : ι → ο
Param ordsucc^ordsucc : ι → ι
Known nat_6^nat_6 : nat_p 6
Known 1c09e.. : ∀ x0 x1 x2 x3 . ∀ x4 : ι → ι → ο . ∀ x5 : ι → ι . equip x0 x1 ⟶ bij x2 x3 x5 ⟶ (∀ x6 : ο . (∀ x7 . and (x7 ⊆ x2) (and (equip x0 x7) (∀ x8 . x8 ∈ x7 ⟶ ∀ x9 . x9 ∈ x7 ⟶ (x8 = x9 ⟶ ∀ x10 : ο . x10) ⟶ x4 (x5 x8) (x5 x9))) ⟶ x6) ⟶ x6) ⟶ ∀ x6 : ο . (∀ x7 . and (x7 ⊆ x3) (and (equip x1 x7) (∀ x8 . x8 ∈ x7 ⟶ ∀ x9 . x9 ∈ x7 ⟶ (x8 = x9 ⟶ ∀ x10 : ο . x10) ⟶ x4 x8 x9)) ⟶ x6) ⟶ x6
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
Known b2ff4.. : ∀ x0 x1 x2 . (x0 = x1 ⟶ ∀ x3 : ο . x3) ⟶ (x0 = x2 ⟶ ∀ x3 : ο . x3) ⟶ (x1 = x2 ⟶ ∀ x3 : ο . x3) ⟶ equip 3 (SetAdjoin (UPair x0 x1) x2)
Known andI^andI : ∀ x0 x1 : ο . x0 ⟶ x1 ⟶ and x0 x1
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 SingE^SingE : ∀ x0 x1 . x1 ∈ Sing x0 ⟶ x1 = x0
Known FalseE^FalseE : False ⟶ ∀ x0 : ο . x0
Theorem ec023.. : ∀ x0 : ι → ι → ο . (∀ x1 x2 . x0 x1 x2 ⟶ x0 x2 x1) ⟶ ∀ x1 . x1 ∈ 6 ⟶ ∀ x2 . x2 ∈ 6 ⟶ ∀ x3 . x3 ∈ 6 ⟶ (x1 = x2 ⟶ ∀ x4 : ο . x4) ⟶ (x1 = x3 ⟶ ∀ x4 : ο . x4) ⟶ (x2 = x3 ⟶ ∀ x4 : ο . x4) ⟶ x0 x1 x2 ⟶ x0 x1 x3 ⟶ x0 x2 x3 ⟶ ∀ x4 : ο . (∀ x5 . and (x5 ⊆ 6) (and (equip 3 x5) (∀ x6 . x6 ∈ x5 ⟶ ∀ x7 . x7 ∈ x5 ⟶ (x6 = x7 ⟶ ∀ x8 : ο . x8) ⟶ x0 x6 x7)) ⟶ x4) ⟶ x4 (proof)
Known xm^xm : ∀ x0 : ο . or x0 (not x0)
Known orIL^orIL : ∀ x0 x1 : ο . x0 ⟶ or x0 x1
Known In_0_4^In_0_4 : 0 ∈ 4
Known In_1_4^In_1_4 : 1 ∈ 4
Known In_0_6^In_0_6 : 0 ∈ 6
Known In_1_6^In_1_6 : 1 ∈ 6
Known In_4_6^In_4_6 : 4 ∈ 6
Known neq_0_1^neq_0_1 : 0 = 1 ⟶ ∀ x0 : ο . x0
Known neq_i_sym^neq_i_sym : ∀ x0 x1 . (x0 = x1 ⟶ ∀ x2 : ο . x2) ⟶ x1 = x0 ⟶ ∀ x2 : ο . x2
Known neq_4_0^neq_4_0 : 4 = 0 ⟶ ∀ x0 : ο . x0
Known neq_4_1^neq_4_1 : 4 = 1 ⟶ ∀ x0 : ο . x0
Known orIR^orIR : ∀ x0 x1 : ο . x1 ⟶ or x0 x1
Known In_5_6^In_5_6 : 5 ∈ 6
Known neq_5_0^neq_5_0 : 5 = 0 ⟶ ∀ x0 : ο . x0
Known neq_5_1^neq_5_1 : 5 = 1 ⟶ ∀ x0 : ο . x0
Known In_2_4^In_2_4 : 2 ∈ 4
Known In_2_6^In_2_6 : 2 ∈ 6
Known neq_0_2^neq_0_2 : 0 = 2 ⟶ ∀ x0 : ο . x0
Known neq_4_2^neq_4_2 : 4 = 2 ⟶ ∀ x0 : ο . x0
Known neq_5_2^neq_5_2 : 5 = 2 ⟶ ∀ x0 : ο . x0
Known In_3_4^In_3_4 : 3 ∈ 4
Known In_3_6^In_3_6 : 3 ∈ 6
Known neq_3_0^neq_3_0 : 3 = 0 ⟶ ∀ x0 : ο . x0
Known neq_4_3^neq_4_3 : 4 = 3 ⟶ ∀ x0 : ο . x0
Known neq_5_3^neq_5_3 : 5 = 3 ⟶ ∀ x0 : ο . x0
Known neq_1_2^neq_1_2 : 1 = 2 ⟶ ∀ x0 : ο . x0
Known neq_3_1^neq_3_1 : 3 = 1 ⟶ ∀ x0 : ο . x0
Known neq_3_2^neq_3_2 : 3 = 2 ⟶ ∀ x0 : ο . x0
Definition nIn^nIn := λ x0 x1 . not (x0 ∈ x1)
Known In_irref^In_irref : ∀ x0 . nIn x0 x0
Known In_no2cycle^In_no2cycle : ∀ x0 x1 . x0 ∈ x1 ⟶ x1 ∈ x0 ⟶ False
Known In_4_5^In_4_5 : 4 ∈ 5
Known neq_5_4^neq_5_4 : 5 = 4 ⟶ ∀ x0 : ο . x0
Known nat_trans^nat_trans : ∀ x0 . nat_p x0 ⟶ ∀ x1 . x1 ∈ x0 ⟶ x1 ⊆ x0
Theorem 245de.. : ∀ x0 : ι → ι → ο . x0 0 4 ⟶ x0 4 5 ⟶ (∀ x1 x2 . x0 x1 x2 ⟶ x0 x2 x1) ⟶ or (∀ x1 : ο . (∀ x2 . and (x2 ⊆ 6) (and (equip 3 x2) (∀ x3 . x3 ∈ x2 ⟶ ∀ x4 . x4 ∈ x2 ⟶ (x3 = x4 ⟶ ∀ x5 : ο . x5) ⟶ x0 x3 x4)) ⟶ x1) ⟶ x1) (∀ x1 : ο . (∀ x2 . and (x2 ⊆ 6) (and (equip 3 x2) (∀ x3 . x3 ∈ x2 ⟶ ∀ x4 . x4 ∈ x2 ⟶ (x3 = x4 ⟶ ∀ x5 : ο . x5) ⟶ not (x0 x3 x4))) ⟶ x1) ⟶ x1) (proof)
Known neq_1_0^neq_1_0 : 1 = 0 ⟶ ∀ x0 : ο . x0
Known cases_6^cases_6 : ∀ x0 . x0 ∈ 6 ⟶ ∀ x1 : ι → ο . x1 0 ⟶ x1 1 ⟶ x1 2 ⟶ x1 3 ⟶ x1 4 ⟶ x1 5 ⟶ x1 x0
Known neq_2_0^neq_2_0 : 2 = 0 ⟶ ∀ x0 : ο . x0
Known neq_2_1^neq_2_1 : 2 = 1 ⟶ ∀ x0 : ο . x0
Param If_i^If_i : ο → ι → ι → ι
Known and3I^and3I : ∀ x0 x1 x2 : ο . x0 ⟶ x1 ⟶ x2 ⟶ and (and x0 x1) x2
Known If_i_1^If_i_1 : ∀ x0 : ο . ∀ x1 x2 . x0 ⟶ If_i x0 x1 x2 = x1
Known If_i_0^If_i_0 : ∀ x0 : ο . ∀ x1 x2 . not x0 ⟶ If_i x0 x1 x2 = x2
Theorem b4917.. : ∀ x0 : ι → ι → ο . x0 4 5 ⟶ (∀ x1 x2 . x0 x1 x2 ⟶ x0 x2 x1) ⟶ or (∀ x1 : ο . (∀ x2 . and (x2 ⊆ 6) (and (equip 3 x2) (∀ x3 . x3 ∈ x2 ⟶ ∀ x4 . x4 ∈ x2 ⟶ (x3 = x4 ⟶ ∀ x5 : ο . x5) ⟶ x0 x3 x4)) ⟶ x1) ⟶ x1) (∀ x1 : ο . (∀ x2 . and (x2 ⊆ 6) (and (equip 3 x2) (∀ x3 . x3 ∈ x2 ⟶ ∀ x4 . x4 ∈ x2 ⟶ (x3 = x4 ⟶ ∀ x5 : ο . x5) ⟶ not (x0 x3 x4))) ⟶ x1) ⟶ x1) (proof)
Known pred_ext_2^pred_ext_2 : ∀ x0 x1 : ι → ο . (∀ x2 . x0 x2 ⟶ x1 x2) ⟶ (∀ x2 . x1 x2 ⟶ x0 x2) ⟶ x0 = x1
Known dneg^dneg : ∀ x0 : ο . not (not x0) ⟶ x0
Theorem TwoRamseyProp_3_3_6^{TwoRamseyProp_3_3_6} : TwoRamseyProp 3 3 6 (proof)