Proofgold Address

Definition and^and := λ x0 x1 : ο . ∀ x2 : ο . (x0 ⟶ x1 ⟶ x2) ⟶ x2
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
Param ordsucc^ordsucc : ι → ι
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 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
Theorem f03aa.. : ∀ x0 . atleastp 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) ⟶ x1) ⟶ x1 (proof)
Definition Subq^Subq := λ x0 x1 . ∀ x2 . x2 ∈ x0 ⟶ x2 ∈ x1
Param If_i^If_i : ο → ι → ι → ι
Param setminus^setminus : ι → ι → ι
Known andI^andI : ∀ x0 x1 : ο . x0 ⟶ x1 ⟶ and x0 x1
Definition or^or := λ x0 x1 : ο . ∀ x2 : ο . (x0 ⟶ x2) ⟶ (x1 ⟶ x2) ⟶ x2
Definition False^False := ∀ x0 : ο . x0
Definition not^not := λ x0 : ο . x0 ⟶ False
Known xm^xm : ∀ x0 : ο . or x0 (not x0)
Known If_i_1^If_i_1 : ∀ x0 : ο . ∀ x1 x2 . x0 ⟶ If_i x0 x1 x2 = x1
Known PowerI^PowerI : ∀ x0 x1 . x1 ⊆ x0 ⟶ x1 ∈ prim4 x0
Definition nIn^nIn := λ x0 x1 . not (x0 ∈ x1)
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 If_i_0^If_i_0 : ∀ x0 : ο . ∀ x1 x2 . not x0 ⟶ If_i x0 x1 x2 = x2
Known set_ext^set_ext : ∀ x0 x1 . x0 ⊆ x1 ⟶ x1 ⊆ x0 ⟶ x0 = x1
Known dneg^dneg : ∀ x0 : ο . not (not x0) ⟶ x0
Known setminusE2^setminusE2 : ∀ x0 x1 x2 . x2 ∈ setminus x0 x1 ⟶ nIn x2 x1
Known setminusI^setminusI : ∀ x0 x1 x2 . x2 ∈ x0 ⟶ nIn x2 x1 ⟶ x2 ∈ setminus x0 x1
Known ordsuccI1^ordsuccI1 : ∀ x0 . x0 ⊆ ordsucc x0
Known PowerE^PowerE : ∀ x0 x1 . x1 ∈ prim4 x0 ⟶ x1 ⊆ x0
Theorem dfb49.. : ∀ x0 x1 . x1 ⊆ prim4 (ordsucc x0) ⟶ (∀ x2 . x2 ∈ x1 ⟶ ∀ x3 . x3 ∈ x1 ⟶ (x2 = x3 ⟶ ∀ x4 : ο . x4) ⟶ ∀ x4 : ο . (∀ x5 . and (x5 ∈ ordsucc x0) (x5 ∈ x2 = x5 ∈ x3) ⟶ x4) ⟶ x4) ⟶ atleastp x1 (prim4 x0) (proof)
Param binunion^binunion : ι → ι → ι
Param setsum^setsum : ι → ι → ι
Param Inj0^Inj0 : ι → ι
Param Inj1^Inj1 : ι → ι
Known binunionE^binunionE : ∀ x0 x1 x2 . x2 ∈ binunion x0 x1 ⟶ or (x2 ∈ x0) (x2 ∈ x1)
Known Inj0_setsum^Inj0_setsum : ∀ x0 x1 x2 . x2 ∈ x0 ⟶ Inj0 x2 ∈ setsum x0 x1
Known Inj1_setsum^Inj1_setsum : ∀ x0 x1 x2 . x2 ∈ x1 ⟶ Inj1 x2 ∈ setsum x0 x1
Known Inj0_inj^Inj0_inj : ∀ x0 x1 . Inj0 x0 = Inj0 x1 ⟶ x0 = x1
Known Inj0_Inj1_neq^{Inj0_Inj1_neq} : ∀ x0 x1 . Inj0 x0 = Inj1 x1 ⟶ ∀ x2 : ο . x2
Known Inj1_inj^Inj1_inj : ∀ x0 x1 . Inj1 x0 = Inj1 x1 ⟶ x0 = x1
Theorem 385ef.. : ∀ x0 x1 x2 x3 . atleastp x0 x2 ⟶ atleastp x1 x3 ⟶ (∀ x4 . x4 ∈ x0 ⟶ nIn x4 x1) ⟶ atleastp (binunion x0 x1) (setsum x2 x3) (proof)
Param nat_p^nat_p : ι → ο
Known PigeonHole_nat^{PigeonHole_nat} : ∀ x0 . nat_p x0 ⟶ ∀ x1 : ι → ι . (∀ x2 . x2 ∈ ordsucc x0 ⟶ x1 x2 ∈ x0) ⟶ not (∀ x2 . x2 ∈ ordsucc x0 ⟶ ∀ x3 . x3 ∈ ordsucc x0 ⟶ x1 x2 = x1 x3 ⟶ x2 = x3)
Theorem 4fb58..^{Pigeonhole_not_atleastp_ordsucc} : ∀ x0 . nat_p x0 ⟶ not (atleastp (ordsucc x0) x0) (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
Param ap^ap : ι → ι → ι
Param lam^Sigma : ι → (ι → ι) → ι
Param Sing^Sing : ι → ι
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
Known cases_4^cases_4 : ∀ x0 . x0 ∈ 4 ⟶ ∀ x1 : ι → ο . x1 0 ⟶ x1 1 ⟶ x1 2 ⟶ x1 3 ⟶ x1 x0
Known tuple_4_0_eq^tuple_4_0_eq : ∀ x0 x1 x2 x3 . ap (lam 4 (λ x5 . If_i (x5 = 0) x0 (If_i (x5 = 1) x1 (If_i (x5 = 2) x2 x3)))) 0 = x0
Known Empty_In_Power^{Empty_In_Power} : ∀ x0 . 0 ∈ prim4 x0
Known tuple_4_1_eq^tuple_4_1_eq : ∀ x0 x1 x2 x3 . ap (lam 4 (λ x5 . If_i (x5 = 0) x0 (If_i (x5 = 1) x1 (If_i (x5 = 2) x2 x3)))) 1 = x1
Known Subq_1_2^Subq_1_2 : 1 ⊆ 2
Known tuple_4_2_eq^tuple_4_2_eq : ∀ x0 x1 x2 x3 . ap (lam 4 (λ x5 . If_i (x5 = 0) x0 (If_i (x5 = 1) x1 (If_i (x5 = 2) x2 x3)))) 2 = x2
Known SingE^SingE : ∀ x0 x1 . x1 ∈ Sing x0 ⟶ x1 = x0
Known In_1_2^In_1_2 : 1 ∈ 2
Known tuple_4_3_eq^tuple_4_3_eq : ∀ x0 x1 x2 x3 . ap (lam 4 (λ x5 . If_i (x5 = 0) x0 (If_i (x5 = 1) x1 (If_i (x5 = 2) x2 x3)))) 3 = x3
Known Self_In_Power^{Self_In_Power} : ∀ x0 . x0 ∈ prim4 x0
Known not_Empty_eq_Sing : ∀ x0 . 0 = Sing x0 ⟶ ∀ x1 : ο . x1
Known neq_1_0^neq_1_0 : 1 = 0 ⟶ ∀ x0 : ο . x0
Known nIn_not_eq_Sing : ∀ x0 x1 . nIn x0 x1 ⟶ x1 = Sing x0 ⟶ ∀ x2 : ο . x2
Known In_irref^In_irref : ∀ x0 . nIn x0 x0
Known In_0_2^In_0_2 : 0 ∈ 2
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 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
Known In_0_4^In_0_4 : 0 ∈ 4
Known In_1_4^In_1_4 : 1 ∈ 4
Known In_2_4^In_2_4 : 2 ∈ 4
Known In_3_4^In_3_4 : 3 ∈ 4
Theorem 26d05.. : equip 4 (prim4 2) (proof)
Known 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
Known Inj1I2^Inj1I2 : ∀ x0 x1 . x1 ∈ x0 ⟶ Inj1 x1 ∈ Inj1 x0
Known Inj1I1^Inj1I1 : ∀ x0 . 0 ∈ Inj1 x0
Known Inj1NE1^Inj1NE1 : ∀ x0 . Inj1 x0 = 0 ⟶ ∀ x1 : ο . x1
Known Inj1E^Inj1E : ∀ x0 x1 . x1 ∈ Inj1 x0 ⟶ or (x1 = 0) (∀ x2 : ο . (∀ x3 . and (x3 ∈ x0) (x1 = Inj1 x3) ⟶ x2) ⟶ x2)
Known ordsuccI2^ordsuccI2 : ∀ x0 . x0 ∈ ordsucc x0
Theorem 4b32b.. : ∀ x0 x1 . equip x0 x1 ⟶ equip (ordsucc x0) (Inj1 x1) (proof)
Known Inj1_setsum_1L^{Inj1_setsum_1L} : ∀ x0 . setsum 1 x0 = Inj1 x0
Theorem bb318.. : equip 5 (setsum 1 (prim4 2)) (proof)
Known nat_5^nat_5 : nat_p 5
Known atleastp_tra^atleastp_tra : ∀ x0 x1 x2 . atleastp x0 x1 ⟶ atleastp x1 x2 ⟶ atleastp x0 x2
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
Theorem 89205.. : ∀ x0 x1 . (∀ x2 . x2 ∈ x0 ⟶ nIn x2 x1) ⟶ atleastp 6 (binunion x0 x1) ⟶ atleastp x0 1 ⟶ atleastp x1 (prim4 2) ⟶ False (proof)
Known cases_3^cases_3 : ∀ x0 . x0 ∈ 3 ⟶ ∀ x1 : ι → ο . x1 0 ⟶ x1 1 ⟶ x1 2 ⟶ x1 x0
Theorem 5de9e.. : ∀ x0 . x0 ⊆ 3 ⟶ ∀ x1 . x1 ⊆ 3 ⟶ 0 ∈ x0 = 0 ∈ x1 ⟶ 1 ∈ x0 = 1 ∈ x1 ⟶ 2 ∈ x0 = 2 ∈ x1 ⟶ x0 = x1 (proof)