Proofgold Asset

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 binunionI1^binunionI1 : ∀ x0 x1 x2 . x2 ∈ x0 ⟶ x2 ∈ binunion x0 x1
Known UPairI1^UPairI1 : ∀ x0 x1 . x0 ∈ UPair x0 x1
Theorem 6be8c.. : ∀ x0 x1 x2 . x0 ∈ SetAdjoin (UPair x0 x1) x2 (proof)
Known UPairI2^UPairI2 : ∀ x0 x1 . x1 ∈ UPair x0 x1
Theorem 535ce.. : ∀ x0 x1 x2 . x1 ∈ SetAdjoin (UPair x0 x1) x2 (proof)
Known binunionI2^binunionI2 : ∀ x0 x1 x2 . x2 ∈ x1 ⟶ x2 ∈ binunion x0 x1
Known SingI^SingI : ∀ x0 . x0 ∈ Sing x0
Theorem f4e2f.. : ∀ x0 x1 x2 . x2 ∈ SetAdjoin (UPair x0 x1) x2 (proof)
Theorem 69a9c.. : ∀ x0 x1 x2 x3 . x0 ∈ SetAdjoin (SetAdjoin (UPair x0 x1) x2) x3 (proof)
Theorem e588e.. : ∀ x0 x1 x2 x3 . x1 ∈ SetAdjoin (SetAdjoin (UPair x0 x1) x2) x3 (proof)
Theorem 14338.. : ∀ x0 x1 x2 x3 . x2 ∈ SetAdjoin (SetAdjoin (UPair x0 x1) x2) x3 (proof)
Theorem b253c.. : ∀ x0 x1 x2 x3 . x3 ∈ SetAdjoin (SetAdjoin (UPair x0 x1) x2) x3 (proof)
Theorem d8272.. : ∀ x0 x1 x2 x3 x4 . x0 ∈ SetAdjoin (SetAdjoin (SetAdjoin (UPair x0 x1) x2) x3) x4 (proof)
Theorem cc191.. : ∀ x0 x1 x2 x3 x4 . x1 ∈ SetAdjoin (SetAdjoin (SetAdjoin (UPair x0 x1) x2) x3) x4 (proof)
Theorem 181b3.. : ∀ x0 x1 x2 x3 x4 . x2 ∈ SetAdjoin (SetAdjoin (SetAdjoin (UPair x0 x1) x2) x3) x4 (proof)
Theorem c9ec0.. : ∀ x0 x1 x2 x3 x4 . x3 ∈ SetAdjoin (SetAdjoin (SetAdjoin (UPair x0 x1) x2) x3) x4 (proof)
Theorem 6143a.. : ∀ x0 x1 x2 x3 x4 . x4 ∈ SetAdjoin (SetAdjoin (SetAdjoin (UPair x0 x1) x2) x3) x4 (proof)
Theorem 3f3e8.. : ∀ x0 x1 x2 x3 x4 x5 . x0 ∈ SetAdjoin (SetAdjoin (SetAdjoin (SetAdjoin (UPair x0 x1) x2) x3) x4) x5 (proof)
Theorem 8ebd9.. : ∀ x0 x1 x2 x3 x4 x5 . x1 ∈ SetAdjoin (SetAdjoin (SetAdjoin (SetAdjoin (UPair x0 x1) x2) x3) x4) x5 (proof)
Theorem 26088.. : ∀ x0 x1 x2 x3 x4 x5 . x2 ∈ SetAdjoin (SetAdjoin (SetAdjoin (SetAdjoin (UPair x0 x1) x2) x3) x4) x5 (proof)
Theorem 6cbbc.. : ∀ x0 x1 x2 x3 x4 x5 . x3 ∈ SetAdjoin (SetAdjoin (SetAdjoin (SetAdjoin (UPair x0 x1) x2) x3) x4) x5 (proof)
Theorem 2fe98.. : ∀ x0 x1 x2 x3 x4 x5 . x4 ∈ SetAdjoin (SetAdjoin (SetAdjoin (SetAdjoin (UPair x0 x1) x2) x3) x4) x5 (proof)
Theorem 9697f.. : ∀ x0 x1 x2 x3 x4 x5 . x5 ∈ SetAdjoin (SetAdjoin (SetAdjoin (SetAdjoin (UPair x0 x1) x2) x3) x4) x5 (proof)
Theorem 600ec.. : ∀ x0 x1 x2 x3 x4 x5 x6 . x0 ∈ SetAdjoin (SetAdjoin (SetAdjoin (SetAdjoin (SetAdjoin (UPair x0 x1) x2) x3) x4) x5) x6 (proof)
Theorem ea770.. : ∀ x0 x1 x2 x3 x4 x5 x6 . x1 ∈ SetAdjoin (SetAdjoin (SetAdjoin (SetAdjoin (SetAdjoin (UPair x0 x1) x2) x3) x4) x5) x6 (proof)
Theorem 651a3.. : ∀ x0 x1 x2 x3 x4 x5 x6 . x2 ∈ SetAdjoin (SetAdjoin (SetAdjoin (SetAdjoin (SetAdjoin (UPair x0 x1) x2) x3) x4) x5) x6 (proof)
Theorem 1b5d7.. : ∀ x0 x1 x2 x3 x4 x5 x6 . x3 ∈ SetAdjoin (SetAdjoin (SetAdjoin (SetAdjoin (SetAdjoin (UPair x0 x1) x2) x3) x4) x5) x6 (proof)
Theorem 44082.. : ∀ x0 x1 x2 x3 x4 x5 x6 . x4 ∈ SetAdjoin (SetAdjoin (SetAdjoin (SetAdjoin (SetAdjoin (UPair x0 x1) x2) x3) x4) x5) x6 (proof)
Theorem 9475f.. : ∀ x0 x1 x2 x3 x4 x5 x6 . x5 ∈ SetAdjoin (SetAdjoin (SetAdjoin (SetAdjoin (SetAdjoin (UPair x0 x1) x2) x3) x4) x5) x6 (proof)
Theorem fe118.. : ∀ x0 x1 x2 x3 x4 x5 x6 . x6 ∈ SetAdjoin (SetAdjoin (SetAdjoin (SetAdjoin (SetAdjoin (UPair x0 x1) x2) x3) x4) x5) x6 (proof)
Theorem 94e7b.. : ∀ x0 x1 x2 x3 x4 x5 x6 x7 . x0 ∈ SetAdjoin (SetAdjoin (SetAdjoin (SetAdjoin (SetAdjoin (SetAdjoin (UPair x0 x1) x2) x3) x4) x5) x6) x7 (proof)
Theorem df755.. : ∀ x0 x1 x2 x3 x4 x5 x6 x7 . x1 ∈ SetAdjoin (SetAdjoin (SetAdjoin (SetAdjoin (SetAdjoin (SetAdjoin (UPair x0 x1) x2) x3) x4) x5) x6) x7 (proof)
Theorem 88aea.. : ∀ x0 x1 x2 x3 x4 x5 x6 x7 . x2 ∈ SetAdjoin (SetAdjoin (SetAdjoin (SetAdjoin (SetAdjoin (SetAdjoin (UPair x0 x1) x2) x3) x4) x5) x6) x7 (proof)
Theorem 2d848.. : ∀ x0 x1 x2 x3 x4 x5 x6 x7 . x3 ∈ SetAdjoin (SetAdjoin (SetAdjoin (SetAdjoin (SetAdjoin (SetAdjoin (UPair x0 x1) x2) x3) x4) x5) x6) x7 (proof)
Theorem 43958.. : ∀ x0 x1 x2 x3 x4 x5 x6 x7 . x4 ∈ SetAdjoin (SetAdjoin (SetAdjoin (SetAdjoin (SetAdjoin (SetAdjoin (UPair x0 x1) x2) x3) x4) x5) x6) x7 (proof)
Theorem e3b9a.. : ∀ x0 x1 x2 x3 x4 x5 x6 x7 . x5 ∈ SetAdjoin (SetAdjoin (SetAdjoin (SetAdjoin (SetAdjoin (SetAdjoin (UPair x0 x1) x2) x3) x4) x5) x6) x7 (proof)
Theorem 696b5.. : ∀ x0 x1 x2 x3 x4 x5 x6 x7 . x6 ∈ SetAdjoin (SetAdjoin (SetAdjoin (SetAdjoin (SetAdjoin (SetAdjoin (UPair x0 x1) x2) x3) x4) x5) x6) x7 (proof)
Theorem 7fdc4.. : ∀ x0 x1 x2 x3 x4 x5 x6 x7 . x7 ∈ SetAdjoin (SetAdjoin (SetAdjoin (SetAdjoin (SetAdjoin (SetAdjoin (UPair x0 x1) x2) x3) x4) x5) x6) x7 (proof)
Param atleastp^atleastp : ι → ι → ο
Param setsum^setsum : ι → ι → ι
Param setminus^setminus : ι → 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 385ef.. : ∀ x0 x1 x2 x3 . atleastp x0 x2 ⟶ atleastp x1 x3 ⟶ (∀ x4 . x4 ∈ x0 ⟶ nIn x4 x1) ⟶ atleastp (binunion x0 x1) (setsum x2 x3)
Known atleastp_tra^atleastp_tra : ∀ x0 x1 x2 . atleastp x0 x1 ⟶ atleastp x1 x2 ⟶ atleastp x0 x2
Definition Subq^Subq := λ x0 x1 . ∀ x2 . x2 ∈ x0 ⟶ x2 ∈ x1
Known Subq_atleastp^{Subq_atleastp} : ∀ x0 x1 . x0 ⊆ x1 ⟶ atleastp x0 x1
Known setminus_Subq^{setminus_Subq} : ∀ x0 x1 . setminus x0 x1 ⊆ x0
Known setminusE2^setminusE2 : ∀ x0 x1 x2 . x2 ∈ setminus x0 x1 ⟶ nIn x2 x1
Known set_ext^set_ext : ∀ x0 x1 . x0 ⊆ x1 ⟶ x1 ⊆ x0 ⟶ x0 = 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 binunionE^binunionE : ∀ x0 x1 x2 . x2 ∈ binunion x0 x1 ⟶ or (x2 ∈ x0) (x2 ∈ x1)
Known FalseE^FalseE : False ⟶ ∀ x0 : ο . x0
Known setminusI^setminusI : ∀ x0 x1 x2 . x2 ∈ x0 ⟶ nIn x2 x1 ⟶ x2 ∈ setminus x0 x1
Known setminusE1^setminusE1 : ∀ x0 x1 x2 . x2 ∈ setminus x0 x1 ⟶ x2 ∈ x0
Theorem c558f.. : ∀ x0 x1 x2 x3 . atleastp x0 x2 ⟶ atleastp x1 x3 ⟶ atleastp (binunion x0 x1) (setsum x2 x3) (proof)
Param ordsucc^ordsucc : ι → ι
Definition u1 := 1
Definition u2 := ordsucc u1
Known cbaf1.. : ∀ x0 x1 . UPair x0 x1 = binunion (Sing x0) (Sing x1)
Known setsum_1_1_2^setsum_1_1_2 : setsum 1 1 = 2
Known 4f2c3.. : ∀ x0 . atleastp (Sing x0) u1
Theorem b3e89.. : ∀ x0 x1 . atleastp (UPair x0 x1) u2 (proof)
Param add_nat^add_nat : ι → ι → ι
Param nat_p^nat_p : ι → ο
Known add_nat_SR^add_nat_SR : ∀ x0 x1 . nat_p x1 ⟶ add_nat x0 (ordsucc x1) = ordsucc (add_nat x0 x1)
Known nat_0^nat_0 : nat_p 0
Known add_nat_0R^add_nat_0R : ∀ x0 . add_nat x0 0 = x0
Theorem f4190.. : ∀ x0 . add_nat x0 u1 = ordsucc x0 (proof)
Definition u3 := ordsucc u2
Param equip^equip : ι → ι → ο
Known equip_atleastp^{equip_atleastp} : ∀ x0 x1 . equip x0 x1 ⟶ atleastp x0 x1
Known equip_sym^equip_sym : ∀ x0 x1 . equip x0 x1 ⟶ equip x1 x0
Known c88e0.. : ∀ x0 x1 x2 x3 . nat_p x0 ⟶ nat_p x1 ⟶ equip x0 x2 ⟶ equip x1 x3 ⟶ equip (add_nat x0 x1) (setsum x2 x3)
Known nat_2^nat_2 : nat_p 2
Known nat_1^nat_1 : nat_p 1
Known equip_ref^equip_ref : ∀ x0 . equip x0 x0
Theorem 51eee.. : ∀ x0 x1 x2 . atleastp (SetAdjoin (UPair x0 x1) x2) u3 (proof)
Definition u4 := ordsucc u3
Known nat_3^nat_3 : nat_p 3
Theorem 38089.. : ∀ x0 x1 x2 x3 . atleastp (SetAdjoin (SetAdjoin (UPair x0 x1) x2) x3) u4 (proof)
Definition u5 := ordsucc u4
Known nat_4^nat_4 : nat_p 4
Theorem 3e701.. : ∀ x0 x1 x2 x3 x4 . atleastp (SetAdjoin (SetAdjoin (SetAdjoin (UPair x0 x1) x2) x3) x4) u5 (proof)
Definition u6 := ordsucc u5
Known nat_5^nat_5 : nat_p 5
Theorem b0421.. : ∀ x0 x1 x2 x3 x4 x5 . atleastp (SetAdjoin (SetAdjoin (SetAdjoin (SetAdjoin (UPair x0 x1) x2) x3) x4) x5) u6 (proof)
Definition u7 := ordsucc u6
Known nat_6^nat_6 : nat_p 6
Theorem 0156c.. : ∀ x0 x1 x2 x3 x4 x5 x6 . atleastp (SetAdjoin (SetAdjoin (SetAdjoin (SetAdjoin (SetAdjoin (UPair x0 x1) x2) x3) x4) x5) x6) u7 (proof)
Definition u8 := ordsucc u7
Known nat_7^nat_7 : nat_p 7
Theorem 657ad.. : ∀ x0 x1 x2 x3 x4 x5 x6 x7 . atleastp (SetAdjoin (SetAdjoin (SetAdjoin (SetAdjoin (SetAdjoin (SetAdjoin (UPair x0 x1) x2) x3) x4) x5) x6) x7) u8 (proof)
Param and^and : ο → ο → ο
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)
Known dneg^dneg : ∀ x0 : ο . not (not x0) ⟶ x0
Definition u9 := ordsucc u8
Known bd9af.. : ∀ x0 : ι → ι → ο . (∀ x1 x2 . x0 x1 x2 ⟶ x0 x2 x1) ⟶ not (or (∀ x1 : ο . (∀ x2 . and (x2 ⊆ u9) (and (equip u3 x2) (∀ x3 . x3 ∈ x2 ⟶ ∀ x4 . x4 ∈ x2 ⟶ (x3 = x4 ⟶ ∀ x5 : ο . x5) ⟶ x0 x3 x4)) ⟶ x1) ⟶ x1) (∀ x1 : ο . (∀ x2 . and (x2 ⊆ u9) (and (equip u4 x2) (∀ x3 . x3 ∈ x2 ⟶ ∀ x4 . x4 ∈ x2 ⟶ (x3 = x4 ⟶ ∀ x5 : ο . x5) ⟶ not (x0 x3 x4))) ⟶ x1) ⟶ x1)) ⟶ ∀ x1 . x1 ∈ u9 ⟶ ∀ x2 : ο . (∀ x3 . x3 ∈ u9 ⟶ ∀ x4 . x4 ∈ u9 ⟶ ∀ x5 . x5 ∈ u9 ⟶ ∀ x6 . x6 ∈ u9 ⟶ (x1 = x3 ⟶ ∀ x7 : ο . x7) ⟶ (x1 = x4 ⟶ ∀ x7 : ο . x7) ⟶ (x1 = x5 ⟶ ∀ x7 : ο . x7) ⟶ (x1 = x6 ⟶ ∀ x7 : ο . x7) ⟶ (x3 = x4 ⟶ ∀ x7 : ο . x7) ⟶ (x3 = x5 ⟶ ∀ x7 : ο . x7) ⟶ (x3 = x6 ⟶ ∀ x7 : ο . x7) ⟶ (x4 = x5 ⟶ ∀ x7 : ο . x7) ⟶ (x4 = x6 ⟶ ∀ x7 : ο . x7) ⟶ (x5 = x6 ⟶ ∀ x7 : ο . x7) ⟶ x0 x1 x3 ⟶ x0 x1 x4 ⟶ x0 x1 x5 ⟶ not (x0 x3 x4) ⟶ not (x0 x3 x5) ⟶ not (x0 x4 x5) ⟶ (∀ x7 . x7 ∈ u9 ⟶ x0 x1 x7 ⟶ x7 ∈ SetAdjoin (SetAdjoin (UPair x1 x3) x4) x5) ⟶ x0 x6 x3 ⟶ x0 x6 x4 ⟶ x2) ⟶ x2
Known 8455a.. : ∀ x0 : ι → ι → ο . (∀ x1 x2 . x0 x1 x2 ⟶ x0 x2 x1) ⟶ not (or (∀ x1 : ο . (∀ x2 . and (x2 ⊆ u9) (and (equip u3 x2) (∀ x3 . x3 ∈ x2 ⟶ ∀ x4 . x4 ∈ x2 ⟶ (x3 = x4 ⟶ ∀ x5 : ο . x5) ⟶ x0 x3 x4)) ⟶ x1) ⟶ x1) (∀ x1 : ο . (∀ x2 . and (x2 ⊆ u9) (and (equip u4 x2) (∀ x3 . x3 ∈ x2 ⟶ ∀ x4 . x4 ∈ x2 ⟶ (x3 = x4 ⟶ ∀ x5 : ο . x5) ⟶ not (x0 x3 x4))) ⟶ x1) ⟶ x1)) ⟶ ∀ x1 . x1 ∈ u9 ⟶ ∀ x2 . x2 ∈ u9 ⟶ ∀ x3 . x3 ∈ u9 ⟶ (x1 = x2 ⟶ ∀ x4 : ο . x4) ⟶ (x1 = x3 ⟶ ∀ x4 : ο . x4) ⟶ (x2 = x3 ⟶ ∀ x4 : ο . x4) ⟶ x0 x1 x2 ⟶ x0 x1 x3 ⟶ ∀ x4 : ο . (∀ x5 . x5 ∈ u9 ⟶ (x1 = x5 ⟶ ∀ x6 : ο . x6) ⟶ (x2 = x5 ⟶ ∀ x6 : ο . x6) ⟶ (x3 = x5 ⟶ ∀ x6 : ο . x6) ⟶ x0 x1 x5 ⟶ not (x0 x2 x5) ⟶ not (x0 x3 x5) ⟶ (∀ x6 . x6 ∈ u9 ⟶ x0 x1 x6 ⟶ x6 ∈ SetAdjoin (SetAdjoin (UPair x1 x2) x3) x5) ⟶ x4) ⟶ x4
Known 1aece.. : ∀ x0 x1 x2 x3 x4 . x4 ∈ SetAdjoin (SetAdjoin (UPair x0 x1) x2) x3 ⟶ ∀ x5 : ι → ο . x5 x0 ⟶ x5 x1 ⟶ x5 x2 ⟶ x5 x3 ⟶ x5 x4
Known c62d8.. : ∀ x0 : ι → ι → ο . (∀ x1 x2 . x0 x1 x2 ⟶ x0 x2 x1) ⟶ not (or (∀ x1 : ο . (∀ x2 . and (x2 ⊆ u9) (and (equip u3 x2) (∀ x3 . x3 ∈ x2 ⟶ ∀ x4 . x4 ∈ x2 ⟶ (x3 = x4 ⟶ ∀ x5 : ο . x5) ⟶ x0 x3 x4)) ⟶ x1) ⟶ x1) (∀ x1 : ο . (∀ x2 . and (x2 ⊆ u9) (and (equip u4 x2) (∀ x3 . x3 ∈ x2 ⟶ ∀ x4 . x4 ∈ x2 ⟶ (x3 = x4 ⟶ ∀ x5 : ο . x5) ⟶ not (x0 x3 x4))) ⟶ x1) ⟶ x1)) ⟶ ∀ x1 . x1 ∈ u9 ⟶ ∀ x2 . x2 ∈ u9 ⟶ ∀ x3 . x3 ∈ u9 ⟶ ∀ x4 . x4 ∈ u9 ⟶ (x1 = x2 ⟶ ∀ x5 : ο . x5) ⟶ (x1 = x3 ⟶ ∀ x5 : ο . x5) ⟶ (x1 = x4 ⟶ ∀ x5 : ο . x5) ⟶ (x2 = x3 ⟶ ∀ x5 : ο . x5) ⟶ (x2 = x4 ⟶ ∀ x5 : ο . x5) ⟶ (x3 = x4 ⟶ ∀ x5 : ο . x5) ⟶ x0 x1 x2 ⟶ x0 x1 x3 ⟶ x0 x1 x4 ⟶ ∀ x5 . x5 ∈ u9 ⟶ x0 x1 x5 ⟶ x5 ∈ SetAdjoin (SetAdjoin (UPair x1 x2) x3) x4
Known neq_i_sym^neq_i_sym : ∀ x0 x1 . (x0 = x1 ⟶ ∀ x2 : ο . x2) ⟶ x1 = x0 ⟶ ∀ x2 : ο . x2
Known 0799b.. : ∀ x0 : ι → ι → ο . (∀ x1 x2 . x0 x1 x2 ⟶ x0 x2 x1) ⟶ not (or (∀ x1 : ο . (∀ x2 . and (x2 ⊆ u9) (and (equip u3 x2) (∀ x3 . x3 ∈ x2 ⟶ ∀ x4 . x4 ∈ x2 ⟶ (x3 = x4 ⟶ ∀ x5 : ο . x5) ⟶ x0 x3 x4)) ⟶ x1) ⟶ x1) (∀ x1 : ο . (∀ x2 . and (x2 ⊆ u9) (and (equip u4 x2) (∀ x3 . x3 ∈ x2 ⟶ ∀ x4 . x4 ∈ x2 ⟶ (x3 = x4 ⟶ ∀ x5 : ο . x5) ⟶ not (x0 x3 x4))) ⟶ x1) ⟶ x1)) ⟶ ∀ x1 . x1 ∈ u9 ⟶ ∀ x2 . x2 ∈ u9 ⟶ ∀ x3 . x3 ∈ u9 ⟶ ∀ x4 . x4 ∈ u9 ⟶ ∀ x5 . x5 ∈ u9 ⟶ (x1 = x2 ⟶ ∀ x6 : ο . x6) ⟶ (x1 = x3 ⟶ ∀ x6 : ο . x6) ⟶ (x1 = x4 ⟶ ∀ x6 : ο . x6) ⟶ (x1 = x5 ⟶ ∀ x6 : ο . x6) ⟶ (x2 = x3 ⟶ ∀ x6 : ο . x6) ⟶ (x2 = x4 ⟶ ∀ x6 : ο . x6) ⟶ (x2 = x5 ⟶ ∀ x6 : ο . x6) ⟶ (x3 = x4 ⟶ ∀ x6 : ο . x6) ⟶ (x3 = x5 ⟶ ∀ x6 : ο . x6) ⟶ (x4 = x5 ⟶ ∀ x6 : ο . x6) ⟶ x0 x1 x2 ⟶ x0 x1 x3 ⟶ x0 x1 x4 ⟶ not (x0 x2 x3) ⟶ not (x0 x2 x4) ⟶ not (x0 x3 x4) ⟶ x0 x5 x2 ⟶ x0 x5 x3 ⟶ x0 x5 x4 ⟶ False
Known 4fb58..^{Pigeonhole_not_atleastp_ordsucc} : ∀ x0 . nat_p x0 ⟶ not (atleastp (ordsucc x0) x0)
Known nat_8^nat_8 : nat_p 8
Known a515c.. : ∀ x0 x1 x2 . (x0 = x1 ⟶ ∀ x3 : ο . x3) ⟶ (x0 = x2 ⟶ ∀ x3 : ο . x3) ⟶ (x1 = x2 ⟶ ∀ x3 : ο . x3) ⟶ equip (SetAdjoin (UPair x0 x1) x2) u3
Known In_0_9^In_0_9 : 0 ∈ 9
Known 0728d.. : ∀ x0 : ι → ι → ο . (∀ x1 x2 . x0 x1 x2 ⟶ x0 x2 x1) ⟶ not (∀ x1 : ο . (∀ x2 . and (x2 ⊆ u9) (and (equip u4 x2) (∀ x3 . x3 ∈ x2 ⟶ ∀ x4 . x4 ∈ x2 ⟶ (x3 = x4 ⟶ ∀ x5 : ο . x5) ⟶ not (x0 x3 x4))) ⟶ x1) ⟶ x1) ⟶ ∀ x1 . x1 ∈ u9 ⟶ ∀ x2 . x2 ∈ u9 ⟶ ∀ x3 . x3 ∈ u9 ⟶ ∀ x4 . x4 ∈ u9 ⟶ not (x0 x1 x2) ⟶ not (x0 x1 x3) ⟶ not (x0 x1 x4) ⟶ not (x0 x2 x3) ⟶ not (x0 x2 x4) ⟶ not (x0 x3 x4) ⟶ ∀ x5 : ο . (x1 = x2 ⟶ x5) ⟶ (x1 = x3 ⟶ x5) ⟶ (x1 = x4 ⟶ x5) ⟶ (x2 = x3 ⟶ x5) ⟶ (x2 = x4 ⟶ x5) ⟶ (x3 = x4 ⟶ x5) ⟶ x5
Known orIR^orIR : ∀ x0 x1 : ο . x1 ⟶ or x0 x1
Theorem TwoRamseyProp_3_4_9^{TwoRamseyProp_3_4_9} : TwoRamseyProp 3 4 9 (proof)