Proofgold Address

Param Sing^Sing : ι → ι
Definition False^False := ∀ x0 : ο . x0
Definition not^not := λ x0 : ο . x0 ⟶ False
Definition nIn^nIn := λ x0 x1 . not (x0 ∈ x1)
Known EmptyE^EmptyE : ∀ x0 . nIn x0 0
Known SingI^SingI : ∀ x0 . x0 ∈ Sing x0
Theorem not_Empty_eq_Sing : ∀ x0 . 0 = Sing x0 ⟶ ∀ x1 : ο . x1 (proof)
Param UPair^UPair : ι → ι → ι
Known UPairI1^UPairI1 : ∀ x0 x1 . x0 ∈ UPair x0 x1
Theorem not_Empty_eq_UPair : ∀ x0 x1 . 0 = UPair x0 x1 ⟶ ∀ x2 : ο . x2 (proof)
Theorem nIn_not_eq_Sing : ∀ x0 x1 . nIn x0 x1 ⟶ x1 = Sing x0 ⟶ ∀ x2 : ο . x2 (proof)
Theorem nIn_not_eq_UPair_1 : ∀ x0 x1 x2 . nIn x0 x2 ⟶ x2 = UPair x0 x1 ⟶ ∀ x3 : ο . x3 (proof)
Known UPairI2^UPairI2 : ∀ x0 x1 . x1 ∈ UPair x0 x1
Theorem nIn_not_eq_UPair_2 : ∀ x0 x1 x2 . nIn x1 x2 ⟶ x2 = UPair x0 x1 ⟶ ∀ x3 : ο . x3 (proof)
Param ordsucc^ordsucc : ι → ι
Param setminus^setminus : ι → ι → ι
Definition Subq^Subq := λ x0 x1 . ∀ x2 . x2 ∈ x0 ⟶ x2 ∈ x1
Definition or^or := λ x0 x1 : ο . ∀ x2 : ο . (x0 ⟶ x2) ⟶ (x1 ⟶ x2) ⟶ x2
Known xm^xm : ∀ x0 : ο . or x0 (not x0)
Known PowerI^PowerI : ∀ x0 x1 . x1 ⊆ x0 ⟶ x1 ∈ prim4 x0
Definition and^and := λ x0 x1 : ο . ∀ x2 : ο . (x0 ⟶ x1 ⟶ x2) ⟶ x2
Known setminusE^setminusE : ∀ x0 x1 x2 . x2 ∈ setminus x0 x1 ⟶ and (x2 ∈ x0) (nIn x2 x1)
Known ordsuccE^ordsuccE : ∀ x0 x1 . x1 ∈ ordsucc x0 ⟶ or (x1 ∈ x0) (x1 = x0)
Known FalseE^FalseE : False ⟶ ∀ x0 : ο . x0
Known PowerE^PowerE : ∀ x0 x1 . x1 ∈ prim4 x0 ⟶ x1 ⊆ x0
Theorem In_Power_ordsucc_cases_impred : ∀ x0 x1 . x1 ∈ prim4 (ordsucc x0) ⟶ ∀ x2 : ο . (x1 ∈ prim4 x0 ⟶ x2) ⟶ (x0 ∈ x1 ⟶ setminus x1 (Sing x0) ∈ prim4 x0 ⟶ x2) ⟶ x2 (proof)
Known Power_0_Sing_0^{Power_0_Sing_0} : prim4 0 = Sing 0
Known SingE^SingE : ∀ x0 x1 . x1 ∈ Sing x0 ⟶ x1 = x0
Theorem In_Power_0_eq_0 : ∀ x0 . x0 ∈ prim4 0 ⟶ x0 = 0 (proof)
Known set_ext^set_ext : ∀ x0 x1 . x0 ⊆ x1 ⟶ x1 ⊆ x0 ⟶ x0 = x1
Known In_0_1^In_0_1 : 0 ∈ 1
Known setminusI^setminusI : ∀ x0 x1 x2 . x2 ∈ x0 ⟶ nIn x2 x1 ⟶ x2 ∈ setminus x0 x1
Known cases_1^cases_1 : ∀ x0 . x0 ∈ 1 ⟶ ∀ x1 : ι → ο . x1 0 ⟶ x1 x0
Theorem In_Power_1_cases_impred : ∀ x0 . x0 ∈ prim4 1 ⟶ ∀ x1 : ο . (x0 = 0 ⟶ x1) ⟶ (x0 = 1 ⟶ x1) ⟶ x1 (proof)
Known dneg^dneg : ∀ x0 : ο . not (not x0) ⟶ x0
Known cases_2^cases_2 : ∀ x0 . x0 ∈ 2 ⟶ ∀ x1 : ι → ο . x1 0 ⟶ x1 1 ⟶ x1 x0
Known setminusE1^setminusE1 : ∀ x0 x1 x2 . x2 ∈ setminus x0 x1 ⟶ x2 ∈ x0
Theorem In_Power_2_cases_impred : ∀ x0 . x0 ∈ prim4 2 ⟶ ∀ x1 : ο . (x0 = 0 ⟶ x1) ⟶ (x0 = 1 ⟶ x1) ⟶ (x0 = Sing 1 ⟶ x1) ⟶ (x0 = 2 ⟶ x1) ⟶ x1 (proof)
Known cases_3^cases_3 : ∀ x0 . x0 ∈ 3 ⟶ ∀ x1 : ι → ο . x1 0 ⟶ x1 1 ⟶ x1 2 ⟶ x1 x0
Known In_irref^In_irref : ∀ x0 . nIn x0 x0
Known neq_1_2^neq_1_2 : 1 = 2 ⟶ ∀ x0 : ο . x0
Known UPairE^UPairE : ∀ x0 x1 x2 . x0 ∈ UPair x1 x2 ⟶ or (x0 = x1) (x0 = 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 In_0_2^In_0_2 : 0 ∈ 2
Known In_1_2^In_1_2 : 1 ∈ 2
Theorem In_Power_3_cases_impred : ∀ x0 . x0 ∈ prim4 3 ⟶ ∀ x1 : ο . (x0 = 0 ⟶ x1) ⟶ (x0 = 1 ⟶ x1) ⟶ (x0 = Sing 1 ⟶ x1) ⟶ (x0 = 2 ⟶ x1) ⟶ (x0 = Sing 2 ⟶ x1) ⟶ (x0 = UPair 0 2 ⟶ x1) ⟶ (x0 = UPair 1 2 ⟶ x1) ⟶ (x0 = 3 ⟶ x1) ⟶ x1 (proof)
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 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)
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 bij_inj^bij_inj : ∀ x0 x1 . ∀ x2 : ι → ι . bij x0 x1 x2 ⟶ inj x0 x1 x2
Theorem equip_atleastp^{equip_atleastp} : ∀ x0 x1 . equip x0 x1 ⟶ atleastp x0 x1 (proof)
Definition TwoRamseyProp_atleastp := λ x0 x1 x2 . ∀ x3 : ι → ι → ο . (∀ x4 x5 . x3 x4 x5 ⟶ x3 x5 x4) ⟶ or (∀ x4 : ο . (∀ x5 . and (x5 ⊆ x2) (and (atleastp 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 (atleastp x1 x5) (∀ x6 . x6 ∈ x5 ⟶ ∀ x7 . x7 ∈ x5 ⟶ (x6 = x7 ⟶ ∀ x8 : ο . x8) ⟶ not (x3 x6 x7))) ⟶ x4) ⟶ x4)
Known andI^andI : ∀ x0 x1 : ο . x0 ⟶ x1 ⟶ and x0 x1
Theorem inj_comp^inj_comp : ∀ x0 x1 x2 . ∀ x3 x4 : ι → ι . inj x0 x1 x3 ⟶ inj x1 x2 x4 ⟶ inj x0 x2 (λ x5 . x4 (x3 x5)) (proof)
Known equip_ref^equip_ref : ∀ x0 . equip x0 x0
Theorem atleastp_ref : ∀ x0 . atleastp x0 x0 (proof)
Theorem atleastp_tra^atleastp_tra : ∀ x0 x1 x2 . atleastp x0 x1 ⟶ atleastp x1 x2 ⟶ atleastp x0 x2 (proof)
Theorem Subq_atleastp^{Subq_atleastp} : ∀ x0 x1 . x0 ⊆ x1 ⟶ atleastp x0 x1 (proof)
Param nat_p^nat_p : ι → ο
Known nat_trans^nat_trans : ∀ x0 . nat_p x0 ⟶ ∀ x1 . x1 ∈ x0 ⟶ x1 ⊆ x0
Theorem nat_In_atleastp : ∀ x0 . nat_p x0 ⟶ ∀ x1 . x1 ∈ x0 ⟶ atleastp x1 x0 (proof)
Known orIL^orIL : ∀ x0 x1 : ο . x0 ⟶ or x0 x1
Known orIR^orIR : ∀ x0 x1 : ο . x1 ⟶ or x0 x1
Theorem TwoRamseyProp_atleastp_atleastp : ∀ x0 x1 x2 x3 x4 . TwoRamseyProp x0 x2 x4 ⟶ atleastp x1 x0 ⟶ atleastp x3 x2 ⟶ TwoRamseyProp_atleastp x1 x3 x4 (proof)