Proofgold Signed Transaction

Definition False^False := ∀ x0 : ο . x0
Definition not^not := λ x0 : ο . x0 ⟶ False
Definition and^and := λ x0 x1 : ο . ∀ x2 : ο . (x0 ⟶ x1 ⟶ x2) ⟶ x2
Definition surj^surj := λ x0 x1 . λ x2 : ι → ι . and (∀ x3 . x3 ∈ x0 ⟶ x2 x3 ∈ x1) (∀ x3 . x3 ∈ x1 ⟶ ∀ x4 : ο . (∀ x5 . and (x5 ∈ x0) (x2 x5 = x3) ⟶ x4) ⟶ x4)
Param Sep^Sep : ι → (ι → ο) → ι
Definition nIn^nIn := λ x0 x1 . not (x0 ∈ x1)
Known Sep_In_Power^Sep_In_Power : ∀ x0 . ∀ x1 : ι → ο . Sep x0 x1 ∈ prim4 x0
Known SepI^SepI : ∀ x0 . ∀ x1 : ι → ο . ∀ x2 . x2 ∈ x0 ⟶ x1 x2 ⟶ x2 ∈ Sep x0 x1
Known SepE2^SepE2 : ∀ x0 . ∀ x1 : ι → ο . ∀ x2 . x2 ∈ Sep x0 x1 ⟶ x1 x2
Theorem feecd..^{form100_63_surjCantor} : ∀ x0 . ∀ x1 : ι → ι . not (surj x0 (prim4 x0) x1) (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)
Param ReplSep^ReplSep : ι → (ι → ο) → (ι → ι) → ι
Known ReplSepI^ReplSepI : ∀ x0 . ∀ x1 : ι → ο . ∀ x2 : ι → ι . ∀ x3 . x3 ∈ x0 ⟶ x1 x3 ⟶ x2 x3 ∈ ReplSep x0 x1 x2
Known ReplSepE_impred^{ReplSepE_impred} : ∀ x0 . ∀ x1 : ι → ο . ∀ x2 : ι → ι . ∀ x3 . x3 ∈ ReplSep x0 x1 x2 ⟶ ∀ x4 : ο . (∀ x5 . x5 ∈ x0 ⟶ x1 x5 ⟶ x3 = x2 x5 ⟶ x4) ⟶ x4
Definition Subq^Subq := λ x0 x1 . ∀ x2 . x2 ∈ x0 ⟶ x2 ∈ x1
Known PowerI^PowerI : ∀ x0 x1 . x1 ⊆ x0 ⟶ x1 ∈ prim4 x0
Theorem form100_63_Cantor^{form100_63_injCantor} : ∀ x0 . ∀ x1 : ι → ι . not (inj (prim4 x0) x0 x1) (proof)
Definition MetaCat_terminal_p^terminal_p := λ x0 : ι → ο . λ x1 : ι → ι → ι → ο . λ x2 : ι → ι . λ x3 : ι → ι → ι → ι → ι → ι . λ x4 . λ x5 : ι → ι . and (x0 x4) (∀ x6 . x0 x6 ⟶ and (x1 x6 x4 (x5 x6)) (∀ x7 . x1 x6 x4 x7 ⟶ x7 = x5 x6))
Param IrreflexiveSymmetricReln^{struct_r_graph} : ι → ο
Param BinRelnHom^Hom_struct_r : ι → ι → ι → ο
Param struct_id^struct_id : ι → ι
Param struct_comp^struct_comp : ι → ι → ι → ι → ι → ι
Param pack_r^pack_r : ι → (ι → ι → ο) → ι
Known 96ca7.. : ∀ x0 . IrreflexiveSymmetricReln x0 ⟶ ∀ x1 : ι → ο . (∀ x2 . ∀ x3 : ι → ι → ο . (∀ x4 . x4 ∈ x2 ⟶ not (x3 x4 x4)) ⟶ (∀ x4 . x4 ∈ x2 ⟶ ∀ x5 . x5 ∈ x2 ⟶ x3 x4 x5 ⟶ x3 x5 x4) ⟶ x1 (pack_r x2 x3)) ⟶ x1 x0
Param Pi^Pi : ι → (ι → ι) → ι
Definition setexp^setexp := λ x0 x1 . Pi x1 (λ x2 . x0)
Param ap^ap : ι → ι → ι
Known c84ab..^{Hom_struct_r_pack} : ∀ x0 x1 . ∀ x2 x3 : ι → ι → ο . ∀ x4 . BinRelnHom (pack_r x0 x2) (pack_r x1 x3) x4 = and (x4 ∈ setexp x1 x0) (∀ x6 . x6 ∈ x0 ⟶ ∀ x7 . x7 ∈ x0 ⟶ x2 x6 x7 ⟶ x3 (ap x4 x6) (ap x4 x7))
Known FalseE^FalseE : False ⟶ ∀ x0 : ο . x0
Known andI^andI : ∀ x0 x1 : ο . x0 ⟶ x1 ⟶ and x0 x1
Known dneg^dneg : ∀ x0 : ο . not (not x0) ⟶ x0
Known ap_Pi^ap_Pi : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 x3 . x2 ∈ Pi x0 x1 ⟶ x3 ∈ x0 ⟶ ap x2 x3 ∈ x1 x3
Known 36176.. : ∀ x0 . ∀ x1 : ι → ι → ο . (∀ x2 . x2 ∈ x0 ⟶ not (x1 x2 x2)) ⟶ (∀ x2 . x2 ∈ x0 ⟶ ∀ x3 . x3 ∈ x0 ⟶ x1 x2 x3 ⟶ x1 x3 x2) ⟶ IrreflexiveSymmetricReln (pack_r x0 x1)
Theorem 24059.. : not (∀ x0 : ο . (∀ x1 . (∀ x2 : ο . (∀ x3 : ι → ι . MetaCat_terminal_p IrreflexiveSymmetricReln BinRelnHom struct_id struct_comp x1 x3 ⟶ x2) ⟶ x2) ⟶ x0) ⟶ x0) (proof)
Param MetaCat_monic_p^monic : (ι → ο) → (ι → ι → ι → ο) → (ι → ι) → (ι → ι → ι → ι → ι → ι) → ι → ι → ι → ο
Param MetaCat_pullback_p^pullback_p : (ι → ο) → (ι → ι → ι → ο) → (ι → ι) → (ι → ι → ι → ι → ι → ι) → ι → ι → ι → ι → ι → ι → ι → ι → (ι → ι → ι → ι) → ο
Definition MetaCat_subobject_classifier_p^{subobject_classifier_p} := λ x0 : ι → ο . λ x1 : ι → ι → ι → ο . λ x2 : ι → ι . λ x3 : ι → ι → ι → ι → ι → ι . λ x4 . λ x5 : ι → ι . λ x6 x7 . λ x8 : ι → ι → ι → ι . λ x9 : ι → ι → ι → ι → ι → ι → ι . and (and (and (MetaCat_terminal_p x0 x1 x2 x3 x4 x5) (x0 x6)) (x1 x4 x6 x7)) (∀ x10 x11 x12 . MetaCat_monic_p x0 x1 x2 x3 x10 x11 x12 ⟶ and (x1 x11 x6 (x8 x10 x11 x12)) (MetaCat_pullback_p x0 x1 x2 x3 x4 x11 x6 x7 (x8 x10 x11 x12) x10 (x5 x10) x12 (x9 x10 x11 x12)))
Theorem 96a7f.. : not (∀ x0 : ο . (∀ x1 . (∀ x2 : ο . (∀ x3 : ι → ι . (∀ x4 : ο . (∀ x5 . (∀ x6 : ο . (∀ x7 . (∀ x8 : ο . (∀ x9 : ι → ι → ι → ι . (∀ x10 : ο . (∀ x11 : ι → ι → ι → ι → ι → ι → ι . MetaCat_subobject_classifier_p IrreflexiveSymmetricReln BinRelnHom struct_id struct_comp x1 x3 x5 x7 x9 x11 ⟶ x10) ⟶ x10) ⟶ x8) ⟶ x8) ⟶ x6) ⟶ x6) ⟶ x4) ⟶ x4) ⟶ x2) ⟶ x2) ⟶ x0) ⟶ x0) (proof)
Definition MetaCat_nno_p^nno_p := λ x0 : ι → ο . λ x1 : ι → ι → ι → ο . λ x2 : ι → ι . λ x3 : ι → ι → ι → ι → ι → ι . λ x4 . λ x5 : ι → ι . λ x6 x7 x8 . λ x9 : ι → ι → ι → ι . and (and (and (and (MetaCat_terminal_p x0 x1 x2 x3 x4 x5) (x0 x6)) (x1 x4 x6 x7)) (x1 x6 x6 x8)) (∀ x10 x11 x12 . x0 x10 ⟶ x1 x4 x10 x11 ⟶ x1 x10 x10 x12 ⟶ and (and (and (x1 x6 x10 (x9 x10 x11 x12)) (x3 x4 x6 x10 (x9 x10 x11 x12) x7 = x11)) (x3 x6 x6 x10 (x9 x10 x11 x12) x8 = x3 x6 x10 x10 x12 (x9 x10 x11 x12))) (∀ x13 . x1 x6 x10 x13 ⟶ x3 x4 x6 x10 x13 x7 = x11 ⟶ x3 x6 x6 x10 x13 x8 = x3 x6 x10 x10 x12 x13 ⟶ x13 = x9 x10 x11 x12))
Theorem b6adc.. : not (∀ x0 : ο . (∀ x1 . (∀ x2 : ο . (∀ x3 : ι → ι . (∀ x4 : ο . (∀ x5 . (∀ x6 : ο . (∀ x7 . (∀ x8 : ο . (∀ x9 . (∀ x10 : ο . (∀ x11 : ι → ι → ι → ι . MetaCat_nno_p IrreflexiveSymmetricReln BinRelnHom struct_id struct_comp x1 x3 x5 x7 x9 x11 ⟶ x10) ⟶ x10) ⟶ x8) ⟶ x8) ⟶ x6) ⟶ x6) ⟶ x4) ⟶ x4) ⟶ x2) ⟶ x2) ⟶ x0) ⟶ x0) (proof)
Param IrreflexiveTransitiveReln^{struct_r_partialord} : ι → ο
Known af4aa.. : ∀ x0 . IrreflexiveTransitiveReln x0 ⟶ ∀ x1 : ι → ο . (∀ x2 . ∀ x3 : ι → ι → ο . (∀ x4 . x4 ∈ x2 ⟶ not (x3 x4 x4)) ⟶ (∀ x4 . x4 ∈ x2 ⟶ ∀ x5 . x5 ∈ x2 ⟶ ∀ x6 . x6 ∈ x2 ⟶ x3 x4 x5 ⟶ x3 x5 x6 ⟶ x3 x4 x6) ⟶ x1 (pack_r x2 x3)) ⟶ x1 x0
Param ZermeloWO^ZermeloWO : ι → ι → ο
Definition ZermeloWOstrict^{ZermeloWOstrict} := λ x0 x1 . and (ZermeloWO x0 x1) (x0 = x1 ⟶ ∀ x2 : ο . x2)
Definition or^or := λ x0 x1 : ο . ∀ x2 : ο . (x0 ⟶ x2) ⟶ (x1 ⟶ x2) ⟶ x2
Definition trichotomous_or^{trichotomous_or} := λ x0 : ι → ι → ο . ∀ x1 x2 . or (or (x0 x1 x2) (x1 = x2)) (x0 x2 x1)
Known ZermeloWOstrict_trich^{ZermeloWOstrict_trich} : trichotomous_or ZermeloWOstrict
Known b25e7.. : ∀ x0 . ∀ x1 : ι → ι → ο . (∀ x2 . x2 ∈ x0 ⟶ not (x1 x2 x2)) ⟶ (∀ x2 . x2 ∈ x0 ⟶ ∀ x3 . x3 ∈ x0 ⟶ ∀ x4 . x4 ∈ x0 ⟶ x1 x2 x3 ⟶ x1 x3 x4 ⟶ x1 x2 x4) ⟶ IrreflexiveTransitiveReln (pack_r x0 x1)
Definition transitive^transitive := λ x0 : ι → ι → ο . ∀ x1 x2 x3 . x0 x1 x2 ⟶ x0 x2 x3 ⟶ x0 x1 x3
Known ZermeloWO_tra^{ZermeloWO_tra} : transitive ZermeloWO
Definition antisymmetric^{antisymmetric} := λ x0 : ι → ι → ο . ∀ x1 x2 . x0 x1 x2 ⟶ x0 x2 x1 ⟶ x1 = x2
Known ZermeloWO_antisym^{ZermeloWO_antisym} : antisymmetric ZermeloWO
Theorem 1db04.. : not (∀ x0 : ο . (∀ x1 . (∀ x2 : ο . (∀ x3 : ι → ι . MetaCat_terminal_p IrreflexiveTransitiveReln BinRelnHom struct_id struct_comp x1 x3 ⟶ x2) ⟶ x2) ⟶ x0) ⟶ x0) (proof)
Theorem f5478.. : not (∀ x0 : ο . (∀ x1 . (∀ x2 : ο . (∀ x3 : ι → ι . (∀ x4 : ο . (∀ x5 . (∀ x6 : ο . (∀ x7 . (∀ x8 : ο . (∀ x9 : ι → ι → ι → ι . (∀ x10 : ο . (∀ x11 : ι → ι → ι → ι → ι → ι → ι . MetaCat_subobject_classifier_p IrreflexiveTransitiveReln BinRelnHom struct_id struct_comp x1 x3 x5 x7 x9 x11 ⟶ x10) ⟶ x10) ⟶ x8) ⟶ x8) ⟶ x6) ⟶ x6) ⟶ x4) ⟶ x4) ⟶ x2) ⟶ x2) ⟶ x0) ⟶ x0) (proof)
Theorem 25a44.. : not (∀ x0 : ο . (∀ x1 . (∀ x2 : ο . (∀ x3 : ι → ι . (∀ x4 : ο . (∀ x5 . (∀ x6 : ο . (∀ x7 . (∀ x8 : ο . (∀ x9 . (∀ x10 : ο . (∀ x11 : ι → ι → ι → ι . MetaCat_nno_p IrreflexiveTransitiveReln BinRelnHom struct_id struct_comp x1 x3 x5 x7 x9 x11 ⟶ x10) ⟶ x10) ⟶ x8) ⟶ x8) ⟶ x6) ⟶ x6) ⟶ x4) ⟶ x4) ⟶ x2) ⟶ x2) ⟶ x0) ⟶ x0) (proof)