Proofgold Address

Definition Subq^Subq := λ x0 x1 . ∀ x2 . x2 ∈ x0 ⟶ x2 ∈ x1
Known set_ext^set_ext : ∀ x0 x1 . x0 ⊆ x1 ⟶ x1 ⊆ x0 ⟶ x0 = x1
Theorem c01ee.. : ∀ x0 x1 . x1 ⊆ x0 ⟶ ∀ x2 . x2 ⊆ x0 ⟶ (∀ x3 . x3 ∈ x0 ⟶ x3 ∈ x1 ⟶ x3 ∈ x2) ⟶ (∀ x3 . x3 ∈ x0 ⟶ x3 ∈ x2 ⟶ x3 ∈ x1) ⟶ x1 = x2 (proof)
Param ordsucc^ordsucc : ι → ι
Definition and^and := λ x0 x1 : ο . ∀ x2 : ο . (x0 ⟶ x1 ⟶ x2) ⟶ x2
Definition or^or := λ x0 x1 : ο . ∀ x2 : ο . (x0 ⟶ x2) ⟶ (x1 ⟶ x2) ⟶ x2
Known cases_3^cases_3 : ∀ x0 . x0 ∈ 3 ⟶ ∀ x1 : ι → ο . x1 0 ⟶ x1 1 ⟶ x1 2 ⟶ x1 x0
Definition False^False := ∀ x0 : ο . x0
Known FalseE^FalseE : False ⟶ ∀ x0 : ο . x0
Known andI^andI : ∀ x0 x1 : ο . x0 ⟶ x1 ⟶ and x0 x1
Known In_2_3^In_2_3 : 2 ∈ 3
Known orIL^orIL : ∀ x0 x1 : ο . x0 ⟶ or x0 x1
Known orIR^orIR : ∀ x0 x1 : ο . x1 ⟶ or x0 x1
Known In_1_3^In_1_3 : 1 ∈ 3
Known In_0_3^In_0_3 : 0 ∈ 3
Theorem 86ed7.. : ∀ x0 . x0 ∈ 3 ⟶ ∀ x1 . x1 ∈ 3 ⟶ (x0 = x1 ⟶ ∀ x2 : ο . x2) ⟶ ∀ x2 : ο . (∀ x3 . and (x3 ∈ 3) (∀ x4 . x4 ∈ 3 ⟶ (x4 = x3 ⟶ ∀ x5 : ο . x5) ⟶ or (x4 = x0) (x4 = x1)) ⟶ x2) ⟶ x2 (proof)
Definition not^not := λ x0 : ο . x0 ⟶ False
Known xm^xm : ∀ x0 : ο . or x0 (not x0)
Known prop_ext_2^prop_ext_2 : ∀ x0 x1 : ο . (x0 ⟶ x1) ⟶ (x1 ⟶ x0) ⟶ x0 = x1
Theorem 29ffa.. : ∀ x0 x1 : ο . (x0 = x1 ⟶ ∀ x2 : ο . x2) ⟶ ∀ x2 : ο . or (x2 = x0) (x2 = 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
Param ap^ap : ι → ι → ι
Param lam^Sigma : ι → (ι → ι) → ι
Param If_i^If_i : ο → ι → 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_2^cases_2 : ∀ x0 . x0 ∈ 2 ⟶ ∀ x1 : ι → ο . x1 0 ⟶ x1 1 ⟶ x1 x0
Known tuple_2_0_eq^tuple_2_0_eq : ∀ x0 x1 . ap (lam 2 (λ x3 . If_i (x3 = 0) x0 x1)) 0 = x0
Known tuple_2_1_eq^tuple_2_1_eq : ∀ x0 x1 . ap (lam 2 (λ x3 . If_i (x3 = 0) x0 x1)) 1 = x1
Known In_0_2^In_0_2 : 0 ∈ 2
Known In_1_2^In_1_2 : 1 ∈ 2
Theorem 1c73f.. : ∀ x0 x1 . x1 ∈ x0 ⟶ ∀ x2 . x2 ∈ x0 ⟶ (x1 = x2 ⟶ ∀ x3 : ο . x3) ⟶ (∀ x3 . x3 ∈ x0 ⟶ or (x3 = x1) (x3 = x2)) ⟶ equip 2 x0 (proof)
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
Definition nIn^nIn := λ x0 x1 . not (x0 ∈ x1)
Known Empty_eq^Empty_eq : ∀ x0 . (∀ x1 . nIn x1 x0) ⟶ x0 = 0
Known EmptyE^EmptyE : ∀ x0 . nIn x0 0
Theorem cb57f.. : ∀ x0 . atleastp x0 0 ⟶ x0 = 0 (proof)
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 c5737.. : ∀ x0 . equip 0 x0 ⟶ x0 = 0 (proof)
Param binunion^binunion : ι → ι → ι
Param setminus^setminus : ι → ι → ι
Param Sing^Sing : ι → ι
Known binunionI2^binunionI2 : ∀ x0 x1 x2 . x2 ∈ x1 ⟶ x2 ∈ binunion x0 x1
Known SingI^SingI : ∀ x0 . x0 ∈ Sing x0
Known binunionI1^binunionI1 : ∀ x0 x1 x2 . x2 ∈ x0 ⟶ x2 ∈ binunion x0 x1
Known setminusI^setminusI : ∀ x0 x1 x2 . x2 ∈ x0 ⟶ nIn x2 x1 ⟶ x2 ∈ setminus x0 x1
Known SingE^SingE : ∀ x0 x1 . x1 ∈ Sing x0 ⟶ x1 = x0
Known binunionE^binunionE : ∀ x0 x1 x2 . x2 ∈ binunion x0 x1 ⟶ or (x2 ∈ x0) (x2 ∈ x1)
Known setminusE1^setminusE1 : ∀ x0 x1 x2 . x2 ∈ setminus x0 x1 ⟶ x2 ∈ x0
Theorem eb0c4..^{binunion_remove1_eq} : ∀ x0 x1 . x1 ∈ x0 ⟶ x0 = binunion (setminus x0 (Sing x1)) (Sing x1) (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 ordsuccE^ordsuccE : ∀ x0 x1 . x1 ∈ ordsucc x0 ⟶ or (x1 ∈ x0) (x1 = x0)
Known If_i_1^If_i_1 : ∀ x0 : ο . ∀ x1 x2 . x0 ⟶ If_i x0 x1 x2 = x1
Known If_i_0^If_i_0 : ∀ x0 : ο . ∀ x1 x2 . not x0 ⟶ If_i x0 x1 x2 = x2
Known In_irref^In_irref : ∀ x0 . nIn x0 x0
Known ordsuccI1^ordsuccI1 : ∀ x0 . x0 ⊆ ordsucc x0
Known ordsuccI2^ordsuccI2 : ∀ x0 . x0 ∈ ordsucc x0
Theorem e8bc0..^{equip_adjoin_ordsucc} : ∀ x0 x1 x2 . nIn x2 x1 ⟶ equip x0 x1 ⟶ equip (ordsucc x0) (binunion x1 (Sing x2)) (proof)
Param setsum^setsum : ι → ι → ι
Param Inj1^Inj1 : ι → ι
Param Inj0^Inj0 : ι → ι
Known setsum_Inj_inv^{setsum_Inj_inv} : ∀ x0 x1 x2 . x2 ∈ setsum x0 x1 ⟶ or (∀ x3 : ο . (∀ x4 . and (x4 ∈ x0) (x2 = Inj0 x4) ⟶ x3) ⟶ x3) (∀ x3 : ο . (∀ x4 . and (x4 ∈ x1) (x2 = Inj1 x4) ⟶ x3) ⟶ x3)
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 ReplI^ReplI : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 . x2 ∈ x0 ⟶ x1 x2 ∈ prim5 x0 x1
Known ReplE_impred^ReplE_impred : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 . x2 ∈ prim5 x0 x1 ⟶ ∀ x3 : ο . (∀ x4 . x4 ∈ x0 ⟶ x2 = x1 x4 ⟶ x3) ⟶ x3
Theorem 020b9.. : ∀ x0 x1 x2 . setsum x0 (binunion x1 x2) = binunion (setsum x0 x1) (prim5 x2 Inj1) (proof)
Param nat_p^nat_p : ι → ο
Param add_nat^add_nat : ι → ι → ι
Known nat_ind^nat_ind : ∀ x0 : ι → ο . x0 0 ⟶ (∀ x1 . nat_p x1 ⟶ x0 x1 ⟶ x0 (ordsucc x1)) ⟶ ∀ x1 . nat_p x1 ⟶ x0 x1
Known add_nat_0R^add_nat_0R : ∀ x0 . add_nat x0 0 = x0
Known equip_tra^equip_tra : ∀ x0 x1 x2 . equip x0 x1 ⟶ equip x1 x2 ⟶ equip x0 x2
Known Inj0_inj^Inj0_inj : ∀ x0 x1 . Inj0 x0 = Inj0 x1 ⟶ x0 = x1
Known add_nat_SR^add_nat_SR : ∀ x0 x1 . nat_p x1 ⟶ add_nat x0 (ordsucc x1) = ordsucc (add_nat x0 x1)
Known Repl_Sing^Repl_Sing : ∀ x0 : ι → ι . ∀ x1 . prim5 (Sing x1) x0 = Sing (x0 x1)
Known Inj0_Inj1_neq^{Inj0_Inj1_neq} : ∀ x0 x1 . Inj0 x0 = Inj1 x1 ⟶ ∀ x2 : ο . x2
Known setminusE2^setminusE2 : ∀ x0 x1 x2 . x2 ∈ setminus x0 x1 ⟶ nIn x2 x1
Known Inj1_inj^Inj1_inj : ∀ x0 x1 . Inj1 x0 = Inj1 x1 ⟶ x0 = x1
Known setminusE^setminusE : ∀ x0 x1 x2 . x2 ∈ setminus x0 x1 ⟶ and (x2 ∈ x0) (nIn x2 x1)
Theorem 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) (proof)
Known nat_0^nat_0 : nat_p 0
Theorem 2eeb3.. : add_nat 2 1 = 3 (proof)
Known nat_1^nat_1 : nat_p 1
Theorem 256ca.. : add_nat 2 2 = 4 (proof)
Known nat_2^nat_2 : nat_p 2
Known equip_ref^equip_ref : ∀ x0 . equip x0 x0
Theorem 5b991.. : equip 4 (setsum 2 2) (proof)
Known PowerE^PowerE : ∀ x0 x1 . x1 ∈ prim4 x0 ⟶ x1 ⊆ x0
Known dneg^dneg : ∀ x0 : ο . not (not x0) ⟶ x0
Known neq_i_sym^neq_i_sym : ∀ x0 x1 . (x0 = x1 ⟶ ∀ x2 : ο . x2) ⟶ x1 = x0 ⟶ ∀ x2 : ο . x2
Known In_0_1^In_0_1 : 0 ∈ 1
Theorem a6047.. : ∀ x0 . x0 ⊆ prim4 3 ⟶ ∀ x1 . x1 ⊆ prim4 3 ⟶ (∀ x2 . x2 ∈ x0 ⟶ ∀ x3 . x3 ∈ x1 ⟶ ∀ x4 : ο . (∀ x5 . and (x5 ∈ 3) (∀ x6 : ο . (∀ x7 . and (x7 ∈ 3) (and (and (x5 = x7 ⟶ ∀ x8 : ο . x8) (x5 ∈ x2 = x5 ∈ x3)) (x7 ∈ x2 = x7 ∈ x3)) ⟶ x6) ⟶ x6) ⟶ x4) ⟶ x4) ⟶ ∀ x2 : ο . (atleastp x0 1 ⟶ x2) ⟶ (atleastp x1 1 ⟶ x2) ⟶ (equip 2 x0 ⟶ equip 2 x1 ⟶ x2) ⟶ x2 (proof)