Proofgold Address

Definition and^and := λ x0 x1 : ο . ∀ x2 : ο . (x0 ⟶ x1 ⟶ x2) ⟶ x2
Definition bij^bij := λ x0 x1 . λ x2 : ι → ι . and (and (∀ 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)
Theorem 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 (proof)
Definition equip^equip := λ x0 x1 . ∀ x2 : ο . (∀ x3 : ι → ι . bij x0 x1 x3 ⟶ x2) ⟶ x2
Param inv^inv : ι → (ι → ι) → ι → ι
Known bij_inv^bij_inv : ∀ x0 x1 . ∀ x2 : ι → ι . bij x0 x1 x2 ⟶ bij x1 x0 (inv x0 x2)
Theorem equip_sym^equip_sym : ∀ x0 x1 . equip x0 x1 ⟶ equip x1 x0 (proof)
Known bij_comp^bij_comp : ∀ x0 x1 x2 . ∀ x3 x4 : ι → ι . bij x0 x1 x3 ⟶ bij x1 x2 x4 ⟶ bij x0 x2 (λ x5 . x4 (x3 x5))
Theorem equip_tra^equip_tra : ∀ x0 x1 x2 . equip x0 x1 ⟶ equip x1 x2 ⟶ equip x0 x2 (proof)
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_ordsucc^nat_ordsucc : ∀ x0 . nat_p x0 ⟶ nat_p (ordsucc x0)
Known nat_2^nat_2 : nat_p 2
Theorem nat_3^nat_3 : nat_p 3 (proof)
Theorem nat_4^nat_4 : nat_p 4 (proof)
Theorem nat_5^nat_5 : nat_p 5 (proof)
Theorem nat_6^nat_6 : nat_p 6 (proof)
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 If_i^If_i : ο → ι → ι → ι
Known and3I^and3I : ∀ x0 x1 x2 : ο . x0 ⟶ x1 ⟶ x2 ⟶ and (and x0 x1) x2
Known cases_3^cases_3 : ∀ x0 . x0 ∈ 3 ⟶ ∀ x1 : ι → ο . x1 0 ⟶ x1 1 ⟶ x1 2 ⟶ x1 x0
Known binunionI1^binunionI1 : ∀ x0 x1 x2 . x2 ∈ x0 ⟶ x2 ∈ binunion x0 x1
Known UPairI1^UPairI1 : ∀ x0 x1 . x0 ∈ UPair x0 x1
Known UPairI2^UPairI2 : ∀ x0 x1 . x1 ∈ UPair x0 x1
Known binunionI2^binunionI2 : ∀ x0 x1 x2 . x2 ∈ x1 ⟶ x2 ∈ binunion x0 x1
Known SingI^SingI : ∀ x0 . x0 ∈ Sing x0
Known FalseE^FalseE : False ⟶ ∀ x0 : ο . x0
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 andI^andI : ∀ x0 x1 : ο . x0 ⟶ x1 ⟶ and x0 x1
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 SingE^SingE : ∀ x0 x1 . x1 ∈ Sing x0 ⟶ x1 = x0
Known If_i_0^If_i_0 : ∀ x0 : ο . ∀ x1 x2 . not x0 ⟶ If_i x0 x1 x2 = x2
Known neq_2_0^neq_2_0 : 2 = 0 ⟶ ∀ x0 : ο . x0
Known neq_2_1^neq_2_1 : 2 = 1 ⟶ ∀ x0 : ο . x0
Known neq_1_0^neq_1_0 : 1 = 0 ⟶ ∀ x0 : ο . x0
Known If_i_1^If_i_1 : ∀ x0 : ο . ∀ x1 x2 . x0 ⟶ If_i x0 x1 x2 = x1
Theorem 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) (proof)
Known and4I^and4I : ∀ x0 x1 x2 x3 : ο . x0 ⟶ x1 ⟶ x2 ⟶ x3 ⟶ and (and (and x0 x1) x2) x3
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_2^neq_1_2 : 1 = 2 ⟶ ∀ x0 : ο . x0
Known or3I1^or3I1 : ∀ x0 x1 x2 : ο . x0 ⟶ or (or x0 x1) x2
Known or3I2^or3I2 : ∀ x0 x1 x2 : ο . x1 ⟶ or (or x0 x1) x2
Known or3I3^or3I3 : ∀ x0 x1 x2 : ο . x2 ⟶ or (or x0 x1) x2
Theorem cf9c7.. : ∀ x0 . equip 3 x0 ⟶ ∀ x1 : ο . (∀ x2 . and (x2 ∈ x0) (∀ x3 : ο . (∀ x4 . and (x4 ∈ x0) (∀ x5 : ο . (∀ x6 . and (x6 ∈ x0) (and (and (and (x2 = x4 ⟶ ∀ x7 : ο . x7) (x2 = x6 ⟶ ∀ x7 : ο . x7)) (x4 = x6 ⟶ ∀ x7 : ο . x7)) (∀ x7 . x7 ∈ x0 ⟶ or (or (x7 = x2) (x7 = x4)) (x7 = x6))) ⟶ x5) ⟶ x5) ⟶ x3) ⟶ x3) ⟶ x1) ⟶ x1 (proof)
Known or3E^or3E : ∀ x0 x1 x2 : ο . or (or x0 x1) x2 ⟶ ∀ x3 : ο . (x0 ⟶ x3) ⟶ (x1 ⟶ x3) ⟶ (x2 ⟶ x3) ⟶ x3
Theorem 7a851.. : ∀ x0 . equip 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) ⟶ (∀ x5 . x5 ∈ x0 ⟶ ∀ x6 : ι → ο . x6 x2 ⟶ x6 x3 ⟶ x6 x4 ⟶ x6 x5) ⟶ x1) ⟶ x1 (proof)
Param ordinal^ordinal : ι → ο
Known ordinal_trichotomy_or^{ordinal_trichotomy_or} : ∀ x0 x1 . ordinal x0 ⟶ ordinal x1 ⟶ or (or (x0 ∈ x1) (x0 = x1)) (x1 ∈ x0)
Theorem f8b84.. : ∀ x0 . equip 3 x0 ⟶ (∀ x1 . x1 ∈ x0 ⟶ ordinal x1) ⟶ ∀ x1 : ο . (∀ x2 . x2 ∈ x0 ⟶ ∀ x3 . x3 ∈ x0 ⟶ ∀ x4 . x4 ∈ x0 ⟶ x2 ∈ x3 ⟶ x3 ∈ x4 ⟶ (∀ x5 . x5 ∈ x0 ⟶ ∀ x6 : ι → ο . x6 x2 ⟶ x6 x3 ⟶ x6 x4 ⟶ x6 x5) ⟶ x1) ⟶ x1 (proof)
Known ReplE_impred^ReplE_impred : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 . x2 ∈ prim5 x0 x1 ⟶ ∀ x3 : ο . (∀ x4 . x4 ∈ x0 ⟶ x2 = x1 x4 ⟶ x3) ⟶ x3
Known ReplI^ReplI : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 . x2 ∈ x0 ⟶ x1 x2 ∈ prim5 x0 x1
Theorem 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 (proof)
Known orIL^orIL : ∀ x0 x1 : ο . x0 ⟶ or x0 x1
Known orIR^orIR : ∀ x0 x1 : ο . x1 ⟶ or x0 x1
Theorem adde3.. : ∀ x0 x1 x2 x3 x4 x5 . equip x0 x3 ⟶ equip x1 x4 ⟶ equip x2 x5 ⟶ TwoRamseyProp x0 x1 x2 ⟶ TwoRamseyProp x3 x4 x5 (proof)