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42aae../ea4c6.. bday: 4947 doc published by Pr6Pc..Definition andand := λ x0 x1 : ο . ∀ x2 : ο . (x0 ⟶ x1 ⟶ x2) ⟶ x2Definition bijbij := 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 bijEbijE : ∀ 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 equipequip := λ x0 x1 . ∀ x2 : ο . (∀ x3 : ι → ι . bij x0 x1 x3 ⟶ x2) ⟶ x2Param invinv : ι → (ι → ι) → ι → ιKnown bij_invbij_inv : ∀ x0 x1 . ∀ x2 : ι → ι . bij x0 x1 x2 ⟶ bij x1 x0 (inv x0 x2)Theorem equip_symequip_sym : ∀ x0 x1 . equip x0 x1 ⟶ equip x1 x0 (proof)Known bij_compbij_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_traequip_tra : ∀ x0 x1 x2 . equip x0 x1 ⟶ equip x1 x2 ⟶ equip x0 x2 (proof)Definition oror := λ x0 x1 : ο . ∀ x2 : ο . (x0 ⟶ x2) ⟶ (x1 ⟶ x2) ⟶ x2Definition SubqSubq := λ x0 x1 . ∀ x2 . x2 ∈ x0 ⟶ x2 ∈ x1Definition FalseFalse := ∀ x0 : ο . x0Definition notnot := λ x0 : ο . x0 ⟶ FalseDefinition TwoRamseyPropTwoRamseyProp := 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_pnat_p : ι → οParam ordsuccordsucc : ι → ιKnown nat_ordsuccnat_ordsucc : ∀ x0 . nat_p x0 ⟶ nat_p (ordsucc x0)Known nat_2nat_2 : nat_p 2Theorem nat_3nat_3 : nat_p 3 (proof)Theorem nat_4nat_4 : nat_p 4 (proof)Theorem nat_5nat_5 : nat_p 5 (proof)Theorem nat_6nat_6 : nat_p 6 (proof)Param binunionbinunion : ι → ι → ιParam SingSing : ι → ιDefinition SetAdjoinSetAdjoin := λ x0 x1 . binunion x0 (Sing x1)Param UPairUPair : ι → ι → ιParam If_iIf_i : ο → ι → ι → ιKnown and3Iand3I : ∀ x0 x1 x2 : ο . x0 ⟶ x1 ⟶ x2 ⟶ and (and x0 x1) x2Known cases_3cases_3 : ∀ x0 . x0 ∈ 3 ⟶ ∀ x1 : ι → ο . x1 0 ⟶ x1 1 ⟶ x1 2 ⟶ x1 x0Known binunionI1binunionI1 : ∀ x0 x1 x2 . x2 ∈ x0 ⟶ x2 ∈ binunion x0 x1Known UPairI1UPairI1 : ∀ x0 x1 . x0 ∈ UPair x0 x1Known UPairI2UPairI2 : ∀ x0 x1 . x1 ∈ UPair x0 x1Known binunionI2binunionI2 : ∀ x0 x1 x2 . x2 ∈ x1 ⟶ x2 ∈ binunion x0 x1Known SingISingI : ∀ x0 . x0 ∈ Sing x0Known FalseEFalseE : False ⟶ ∀ x0 : ο . x0Known binunionEbinunionE : ∀ x0 x1 x2 . x2 ∈ binunion x0 x1 ⟶ or (x2 ∈ x0) (x2 ∈ x1)Known UPairEUPairE : ∀ x0 x1 x2 . x0 ∈ UPair x1 x2 ⟶ or (x0 = x1) (x0 = x2)Known andIandI : ∀ x0 x1 : ο . x0 ⟶ x1 ⟶ and x0 x1Known In_0_3In_0_3 : 0 ∈ 3Known In_1_3In_1_3 : 1 ∈ 3Known In_2_3In_2_3 : 2 ∈ 3Known SingESingE : ∀ x0 x1 . x1 ∈ Sing x0 ⟶ x1 = x0Known If_i_0If_i_0 : ∀ x0 : ο . ∀ x1 x2 . not x0 ⟶ If_i x0 x1 x2 = x2Known neq_2_0neq_2_0 : 2 = 0 ⟶ ∀ x0 : ο . x0Known neq_2_1neq_2_1 : 2 = 1 ⟶ ∀ x0 : ο . x0Known neq_1_0neq_1_0 : 1 = 0 ⟶ ∀ x0 : ο . x0Known If_i_1If_i_1 : ∀ x0 : ο . ∀ x1 x2 . x0 ⟶ If_i x0 x1 x2 = x1Theorem 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 and4Iand4I : ∀ x0 x1 x2 x3 : ο . x0 ⟶ x1 ⟶ x2 ⟶ x3 ⟶ and (and (and x0 x1) x2) x3Known neq_0_1neq_0_1 : 0 = 1 ⟶ ∀ x0 : ο . x0Known neq_0_2neq_0_2 : 0 = 2 ⟶ ∀ x0 : ο . x0Known neq_1_2neq_1_2 : 1 = 2 ⟶ ∀ x0 : ο . x0Known or3I1or3I1 : ∀ x0 x1 x2 : ο . x0 ⟶ or (or x0 x1) x2Known or3I2or3I2 : ∀ x0 x1 x2 : ο . x1 ⟶ or (or x0 x1) x2Known or3I3or3I3 : ∀ x0 x1 x2 : ο . x2 ⟶ or (or x0 x1) x2Theorem 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 or3Eor3E : ∀ x0 x1 x2 : ο . or (or x0 x1) x2 ⟶ ∀ x3 : ο . (x0 ⟶ x3) ⟶ (x1 ⟶ x3) ⟶ (x2 ⟶ x3) ⟶ x3Theorem 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 ordinalordinal : ι → οKnown ordinal_trichotomy_orordinal_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_impredReplE_impred : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 . x2 ∈ prim5 x0 x1 ⟶ ∀ x3 : ο . (∀ x4 . x4 ∈ x0 ⟶ x2 = x1 x4 ⟶ x3) ⟶ x3Known ReplIReplI : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 . x2 ∈ x0 ⟶ x1 x2 ∈ prim5 x0 x1Theorem 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 orILorIL : ∀ x0 x1 : ο . x0 ⟶ or x0 x1Known orIRorIR : ∀ x0 x1 : ο . x1 ⟶ or x0 x1Theorem 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)
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