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
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
Param equip^equip : ι → ι → ο
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_5^nat_5 : nat_p 5
Param ordinal^ordinal : ι → ο
Known 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
Known FalseE^FalseE : False ⟶ ∀ 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 neq_4_1^neq_4_1 : 4 = 1 ⟶ ∀ x0 : ο . x0
Known neq_4_3^neq_4_3 : 4 = 3 ⟶ ∀ x0 : ο . x0
Known neq_3_2^neq_3_2 : 3 = 2 ⟶ ∀ x0 : ο . x0
Known neq_4_2^neq_4_2 : 4 = 2 ⟶ ∀ x0 : ο . x0
Known neq_0_1^neq_0_1 : 0 = 1 ⟶ ∀ x0 : ο . x0
Known neq_4_0^neq_4_0 : 4 = 0 ⟶ ∀ x0 : ο . x0
Known neq_2_1^neq_2_1 : 2 = 1 ⟶ ∀ x0 : ο . x0
Known neq_3_1^neq_3_1 : 3 = 1 ⟶ ∀ x0 : ο . x0
Known neq_3_0^neq_3_0 : 3 = 0 ⟶ ∀ x0 : ο . x0
Known neq_1_0^neq_1_0 : 1 = 0 ⟶ ∀ x0 : ο . x0
Known neq_2_0^neq_2_0 : 2 = 0 ⟶ ∀ x0 : ο . x0
Known or5E^or5E : ∀ x0 x1 x2 x3 x4 : ο . or (or (or (or x0 x1) x2) x3) x4 ⟶ ∀ x5 : ο . (x0 ⟶ x5) ⟶ (x1 ⟶ x5) ⟶ (x2 ⟶ x5) ⟶ (x3 ⟶ x5) ⟶ (x4 ⟶ x5) ⟶ x5
Known or5I5^or5I5 : ∀ x0 x1 x2 x3 x4 : ο . x4 ⟶ or (or (or (or x0 x1) x2) x3) x4
Known andI^andI : ∀ x0 x1 : ο . x0 ⟶ x1 ⟶ and x0 x1
Known orIR^orIR : ∀ x0 x1 : ο . x1 ⟶ or x0 x1
Known orIL^orIL : ∀ x0 x1 : ο . x0 ⟶ or x0 x1
Known or5I4^or5I4 : ∀ x0 x1 x2 x3 x4 : ο . x3 ⟶ or (or (or (or x0 x1) x2) x3) x4
Known or5I3^or5I3 : ∀ x0 x1 x2 x3 x4 : ο . x2 ⟶ or (or (or (or x0 x1) x2) x3) x4
Known or5I2^or5I2 : ∀ x0 x1 x2 x3 x4 : ο . x1 ⟶ or (or (or (or x0 x1) x2) x3) x4
Known or5I1^or5I1 : ∀ x0 x1 x2 x3 x4 : ο . x0 ⟶ or (or (or (or x0 x1) x2) x3) x4
Theorem e6ef8.. : ∀ x0 : ο . (∀ x1 : ι → ι → ο . (∀ x2 : ο . ((∀ x3 x4 . x1 x3 x4 ⟶ x1 x4 x3) ⟶ x1 0 1 ⟶ x1 1 2 ⟶ x1 2 3 ⟶ x1 3 4 ⟶ x1 4 0 ⟶ not (x1 0 2) ⟶ not (x1 0 3) ⟶ not (x1 1 3) ⟶ not (x1 1 4) ⟶ not (x1 2 4) ⟶ x2) ⟶ x2) ⟶ x0) ⟶ x0 (proof)
Known cases_5^cases_5 : ∀ x0 . x0 ∈ 5 ⟶ ∀ x1 : ι → ο . x1 0 ⟶ x1 1 ⟶ x1 2 ⟶ x1 3 ⟶ x1 4 ⟶ x1 x0
Definition nIn^nIn := λ x0 x1 . not (x0 ∈ x1)
Known EmptyE^EmptyE : ∀ x0 . nIn x0 0
Known cases_1^cases_1 : ∀ x0 . x0 ∈ 1 ⟶ ∀ x1 : ι → ο . x1 0 ⟶ x1 x0
Known cases_2^cases_2 : ∀ x0 . x0 ∈ 2 ⟶ ∀ x1 : ι → ο . x1 0 ⟶ x1 1 ⟶ x1 x0
Known cases_3^cases_3 : ∀ x0 . x0 ∈ 3 ⟶ ∀ x1 : ι → ο . x1 0 ⟶ x1 1 ⟶ x1 2 ⟶ x1 x0
Known neq_i_sym^neq_i_sym : ∀ x0 x1 . (x0 = x1 ⟶ ∀ x2 : ο . x2) ⟶ x1 = x0 ⟶ ∀ x2 : ο . x2
Known cases_4^cases_4 : ∀ x0 . x0 ∈ 4 ⟶ ∀ x1 : ι → ο . x1 0 ⟶ x1 1 ⟶ x1 2 ⟶ x1 3 ⟶ x1 x0
Known nat_p_ordinal^{nat_p_ordinal} : ∀ x0 . nat_p x0 ⟶ ordinal x0
Known nat_p_trans^nat_p_trans : ∀ x0 . nat_p x0 ⟶ ∀ x1 . x1 ∈ x0 ⟶ nat_p x1
Theorem not_TwoRamseyProp_3_3_5^{not_TwoRamseyProp_3_3_5} : not (TwoRamseyProp 3 3 5) (proof)