Proofgold Address

Definition False^False := ∀ x0 : ο . x0
Definition not^not := λ x0 : ο . x0 ⟶ False
Param surj^surj : ι → ι → (ι → ι) → ο
Param omega^omega : ι
Known form100_22_v3^{form100_22_v3} : ∀ x0 : ι → ι . not (surj omega (prim4 omega) x0)
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
Known bij_surj^bij_surj : ∀ x0 x1 . ∀ x2 : ι → ι . bij x0 x1 x2 ⟶ surj x0 x1 x2
Theorem form100_22_v1^{form100_22_v1} : not (equip omega (prim4 omega)) (proof)
Param SNoLt^SNoLt : ι → ι → ο
Param eps_^eps_ : ι → ι
Param ordsucc^ordsucc : ι → ι
Param nat_p^nat_p : ι → ο
Param add_SNo^add_SNo : ι → ι → ι
Known eps_ordsucc_half_add^{eps_ordsucc_half_add} : ∀ x0 . nat_p x0 ⟶ add_SNo (eps_ (ordsucc x0)) (eps_ (ordsucc x0)) = eps_ x0
Known omega_nat_p^omega_nat_p : ∀ x0 . x0 ∈ omega ⟶ nat_p x0
Param SNo^SNo : ι → ο
Known add_SNo_eps_Lt^{add_SNo_eps_Lt} : ∀ x0 . SNo x0 ⟶ ∀ x1 . x1 ∈ omega ⟶ SNoLt x0 (add_SNo x0 (eps_ x1))
Known SNo_eps_^SNo_eps_ : ∀ x0 . x0 ∈ omega ⟶ SNo (eps_ x0)
Known omega_ordsucc^{omega_ordsucc} : ∀ x0 . x0 ∈ omega ⟶ ordsucc x0 ∈ omega
Theorem eps_ordsucc_Lt : ∀ x0 . x0 ∈ omega ⟶ SNoLt (eps_ (ordsucc x0)) (eps_ x0) (proof)
Param real^real : ι
Definition and^and := λ x0 x1 : ο . ∀ x2 : ο . (x0 ⟶ x1 ⟶ x2) ⟶ x2
Known dneg^dneg : ∀ x0 : ο . not (not x0) ⟶ x0
Param SNoLev^SNoLev : ι → ι
Param SNoS_^SNoS_ : ι → ι
Param minus_SNo^minus_SNo : ι → ι
Param abs_SNo^abs_SNo : ι → ι
Known real_E^real_E : ∀ x0 . x0 ∈ real ⟶ ∀ x1 : ο . (SNo x0 ⟶ SNoLev x0 ∈ ordsucc omega ⟶ x0 ∈ SNoS_ (ordsucc omega) ⟶ SNoLt (minus_SNo omega) x0 ⟶ SNoLt x0 omega ⟶ (∀ x2 . x2 ∈ SNoS_ omega ⟶ (∀ x3 . x3 ∈ omega ⟶ SNoLt (abs_SNo (add_SNo x2 (minus_SNo x0))) (eps_ x3)) ⟶ x2 = x0) ⟶ (∀ x2 . x2 ∈ omega ⟶ ∀ x3 : ο . (∀ x4 . and (x4 ∈ SNoS_ omega) (and (SNoLt x4 x0) (SNoLt x0 (add_SNo x4 (eps_ x2)))) ⟶ x3) ⟶ x3) ⟶ x1) ⟶ x1
Known SNoLt_irref^SNoLt_irref : ∀ x0 . not (SNoLt x0 x0)
Param ordinal^ordinal : ι → ο
Definition Subq^Subq := λ x0 x1 . ∀ x2 . x2 ∈ x0 ⟶ x2 ∈ x1
Param binunion^binunion : ι → ι → ι
Param SetAdjoin^SetAdjoin : ι → ι → ι
Param Sing^Sing : ι → ι
Definition SNoElts_^SNoElts_ := λ x0 . binunion x0 {SetAdjoin x1 (Sing 1)|x1 ∈ x0}
Param exactly1of2^exactly1of2 : ο → ο → ο
Definition SNo_^SNo_ := λ x0 x1 . and (x1 ⊆ SNoElts_ x0) (∀ x2 . x2 ∈ x0 ⟶ exactly1of2 (SetAdjoin x2 (Sing 1) ∈ x1) (x2 ∈ x1))
Known SNoS_I^SNoS_I : ∀ x0 . ordinal x0 ⟶ ∀ x1 x2 . x2 ∈ x0 ⟶ SNo_ x2 x1 ⟶ x1 ∈ SNoS_ x0
Known omega_ordinal^{omega_ordinal} : ordinal omega
Known nat_p_omega^nat_p_omega : ∀ x0 . nat_p x0 ⟶ x0 ∈ omega
Known nat_0^nat_0 : nat_p 0
Known ordinal_SNo_^ordinal_SNo_ : ∀ x0 . ordinal x0 ⟶ SNo_ x0 x0
Known ordinal_Empty^{ordinal_Empty} : ordinal 0
Known add_SNo_0L^add_SNo_0L : ∀ x0 . SNo x0 ⟶ add_SNo 0 x0 = x0
Known SNo_minus_SNo^{SNo_minus_SNo} : ∀ x0 . SNo x0 ⟶ SNo (minus_SNo x0)
Known abs_SNo_minus^{abs_SNo_minus} : ∀ x0 . SNo x0 ⟶ abs_SNo (minus_SNo x0) = abs_SNo x0
Param SNoLe^SNoLe : ι → ι → ο
Known nonneg_abs_SNo^{nonneg_abs_SNo} : ∀ x0 . SNoLe 0 x0 ⟶ abs_SNo x0 = x0
Known SNoLtLe^SNoLtLe : ∀ x0 x1 . SNoLt x0 x1 ⟶ SNoLe x0 x1
Definition or^or := λ x0 x1 : ο . ∀ x2 : ο . (x0 ⟶ x2) ⟶ (x1 ⟶ x2) ⟶ x2
Known SNoLtLe_or^SNoLtLe_or : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ or (SNoLt x0 x1) (SNoLe x1 x0)
Known FalseE^FalseE : False ⟶ ∀ x0 : ο . x0
Known andI^andI : ∀ x0 x1 : ο . x0 ⟶ x1 ⟶ and x0 x1
Known SNoLtLe_tra^SNoLtLe_tra : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNoLt x0 x1 ⟶ SNoLe x1 x2 ⟶ SNoLt x0 x2
Theorem real_pos_eps_ : ∀ x0 . x0 ∈ real ⟶ SNoLt 0 x0 ⟶ ∀ x1 : ο . (∀ x2 . and (x2 ∈ omega) (SNoLt (eps_ x2) x0) ⟶ x1) ⟶ x1 (proof)
Known real_add_SNo^real_add_SNo : ∀ x0 . x0 ∈ real ⟶ ∀ x1 . x1 ∈ real ⟶ add_SNo x0 x1 ∈ real
Known real_minus_SNo^{real_minus_SNo} : ∀ x0 . x0 ∈ real ⟶ minus_SNo x0 ∈ real
Known add_SNo_minus_Lt2b^{add_SNo_minus_Lt2b} : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNoLt (add_SNo x2 x1) x0 ⟶ SNoLt x2 (add_SNo x0 (minus_SNo x1))
Known SNo_0^SNo_0 : SNo 0
Known SNoS_E2^SNoS_E2 : ∀ x0 . ordinal x0 ⟶ ∀ x1 . x1 ∈ SNoS_ x0 ⟶ ∀ x2 : ο . (SNoLev x1 ∈ x0 ⟶ ordinal (SNoLev x1) ⟶ SNo x1 ⟶ SNo_ (SNoLev x1) x1 ⟶ x2) ⟶ x2
Known add_SNo_minus_Lt2^{add_SNo_minus_Lt2} : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNoLt x2 (add_SNo x0 (minus_SNo x1)) ⟶ SNoLt (add_SNo x2 x1) x0
Known add_SNo_Lt2_cancel^{add_SNo_Lt2_cancel} : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNoLt (add_SNo x0 x1) (add_SNo x0 x2) ⟶ SNoLt x1 x2
Known SNo_add_SNo^SNo_add_SNo : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNo (add_SNo x0 x1)
Known add_SNo_0R^add_SNo_0R : ∀ x0 . SNo x0 ⟶ add_SNo x0 0 = x0
Known add_SNo_com_3_0_1^{add_SNo_com_3_0_1} : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ add_SNo x0 (add_SNo x1 x2) = add_SNo x1 (add_SNo x0 x2)
Known SNoLt_tra^SNoLt_tra : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNoLt x0 x1 ⟶ SNoLt x1 x2 ⟶ SNoLt x0 x2
Known SNo_add_SNo_3^{SNo_add_SNo_3} : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNo (add_SNo x0 (add_SNo x1 x2))
Known add_SNo_Lt2^add_SNo_Lt2 : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNoLt x1 x2 ⟶ SNoLt (add_SNo x0 x1) (add_SNo x0 x2)
Known real_SNo^real_SNo : ∀ x0 . x0 ∈ real ⟶ SNo x0
Theorem real_Lt_SNoS_omega_inter : ∀ x0 . x0 ∈ real ⟶ ∀ x1 . x1 ∈ real ⟶ SNoLt x0 x1 ⟶ ∀ x2 : ο . (∀ x3 . and (x3 ∈ SNoS_ omega) (and (SNoLt x0 x3) (SNoLt x3 x1)) ⟶ x2) ⟶ x2 (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
Param ReplSep^ReplSep : ι → (ι → ο) → (ι → ι) → ι
Param mul_SNo^mul_SNo : ι → ι → ι
Known PowerI^PowerI : ∀ x0 x1 . x1 ⊆ x0 ⟶ x1 ∈ prim4 x0
Known binunion_Subq_min^{binunion_Subq_min} : ∀ x0 x1 x2 . x0 ⊆ x2 ⟶ x1 ⊆ x2 ⟶ binunion 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
Known mul_SNo_In_omega^{mul_SNo_In_omega} : ∀ x0 . x0 ∈ omega ⟶ ∀ x1 . x1 ∈ omega ⟶ mul_SNo x0 x1 ∈ omega
Known nat_2^nat_2 : nat_p 2
Known add_SNo_In_omega^{add_SNo_In_omega} : ∀ x0 . x0 ∈ omega ⟶ ∀ x1 . x1 ∈ omega ⟶ add_SNo x0 x1 ∈ omega
Known nat_1^nat_1 : nat_p 1
Known ordsucc_omega_ordinal^{ordsucc_omega_ordinal} : ordinal (ordsucc omega)
Known set_ext^set_ext : ∀ x0 x1 . x0 ⊆ x1 ⟶ x1 ⊆ x0 ⟶ x0 = x1
Param TransSet^TransSet : ι → ο
Known TransSet_In_ordsucc_Subq^{TransSet_In_ordsucc_Subq} : ∀ x0 x1 . TransSet x1 ⟶ x0 ∈ ordsucc x1 ⟶ x0 ⊆ x1
Known omega_TransSet^{omega_TransSet} : TransSet omega
Known binunionE^binunionE : ∀ x0 x1 x2 . x2 ∈ binunion x0 x1 ⟶ or (x2 ∈ x0) (x2 ∈ x1)
Known binunionI1^binunionI1 : ∀ x0 x1 x2 . x2 ∈ x0 ⟶ x2 ∈ binunion x0 x1
Known ReplSepI^ReplSepI : ∀ x0 . ∀ x1 : ι → ο . ∀ x2 : ι → ι . ∀ x3 . x3 ∈ x0 ⟶ x1 x3 ⟶ x2 x3 ∈ ReplSep x0 x1 x2
Known mul_SNo_nonzero_cancel^{mul_SNo_nonzero_cancel_L} : ∀ x0 x1 x2 . SNo x0 ⟶ (x0 = 0 ⟶ ∀ x3 : ο . x3) ⟶ SNo x1 ⟶ SNo x2 ⟶ mul_SNo x0 x1 = mul_SNo x0 x2 ⟶ x1 = x2
Known SNo_2^SNo_2 : SNo 2
Known neq_2_0^neq_2_0 : 2 = 0 ⟶ ∀ x0 : ο . x0
Known omega_SNo^omega_SNo : ∀ x0 . x0 ∈ omega ⟶ SNo x0
Param even_nat^even_nat : ι → ο
Param odd_nat^odd_nat : ι → ο
Known even_nat_not_odd_nat^{even_nat_not_odd_nat} : ∀ x0 . even_nat x0 ⟶ not (odd_nat x0)
Param mul_nat^mul_nat : ι → ι → ι
Known mul_nat_mul_SNo^{mul_nat_mul_SNo} : ∀ x0 . x0 ∈ omega ⟶ ∀ x1 . x1 ∈ omega ⟶ mul_nat x0 x1 = mul_SNo x0 x1
Known even_nat_double^{even_nat_double} : ∀ x0 . nat_p x0 ⟶ even_nat (mul_nat 2 x0)
Known add_SNo_com^add_SNo_com : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ add_SNo x0 x1 = add_SNo x1 x0
Known SNo_1^SNo_1 : SNo 1
Known ordinal_ordsucc_SNo_eq^{ordinal_ordsucc_SNo_eq} : ∀ x0 . ordinal x0 ⟶ ordsucc x0 = add_SNo 1 x0
Known nat_p_ordinal^{nat_p_ordinal} : ∀ x0 . nat_p x0 ⟶ ordinal x0
Known even_nat_odd_nat_S^{even_nat_odd_nat_S} : ∀ x0 . even_nat x0 ⟶ odd_nat (ordsucc x0)
Known ReplE_impred^ReplE_impred : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 . x2 ∈ prim5 x0 x1 ⟶ ∀ x3 : ο . (∀ x4 . x4 ∈ x0 ⟶ x2 = x1 x4 ⟶ x3) ⟶ x3
Known binunionI2^binunionI2 : ∀ x0 x1 x2 . x2 ∈ x1 ⟶ x2 ∈ binunion x0 x1
Known add_SNo_cancel_R^{add_SNo_cancel_R} : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ add_SNo x0 x1 = add_SNo x2 x1 ⟶ x0 = x2
Known SNo_mul_SNo^SNo_mul_SNo : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNo (mul_SNo x0 x1)
Theorem atleastp_SNoS_ordsucc_omega_Power_omega^{atleastp_SNoS_ordsucc_omega_Power_omega} : atleastp (SNoS_ (ordsucc omega)) (prim4 omega) (proof)
Param finite^finite : ι → ο
Definition infinite^infinite := λ x0 . not (finite x0)
Param nIn^nIn : ι → ι → ο
Known finite_ind^finite_ind : ∀ x0 : ι → ο . x0 0 ⟶ (∀ x1 x2 . finite x1 ⟶ nIn x2 x1 ⟶ x0 x1 ⟶ x0 (binunion x1 (Sing x2))) ⟶ ∀ x1 . finite x1 ⟶ x0 x1
Known Repl_Empty^Repl_Empty : ∀ x0 : ι → ι . prim5 0 x0 = 0
Known finite_Empty^finite_Empty : finite 0
Known ReplI^ReplI : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 . x2 ∈ x0 ⟶ x1 x2 ∈ prim5 x0 x1
Known SingE^SingE : ∀ x0 x1 . x1 ∈ Sing x0 ⟶ x1 = x0
Known SingI^SingI : ∀ x0 . x0 ∈ Sing x0
Known binunion_finite^{binunion_finite} : ∀ x0 . finite x0 ⟶ ∀ x1 . finite x1 ⟶ finite (binunion x0 x1)
Known 28148..^Sing_finite : ∀ x0 . finite (Sing x0)
Theorem Repl_finite^Repl_finite : ∀ x0 : ι → ι . ∀ x1 . finite x1 ⟶ finite (prim5 x1 x0) (proof)
Known Subq_finite^Subq_finite : ∀ x0 . finite x0 ⟶ ∀ x1 . x1 ⊆ x0 ⟶ finite x1
Known 9b83b..^nat_finite : ∀ x0 . nat_p x0 ⟶ finite x0
Known nat_ordsucc^nat_ordsucc : ∀ x0 . nat_p x0 ⟶ nat_p (ordsucc x0)
Known ordinal_trichotomy_or_impred^{ordinal_trichotomy_or_impred} : ∀ x0 x1 . ordinal x0 ⟶ ordinal x1 ⟶ ∀ x2 : ο . (x0 ∈ x1 ⟶ x2) ⟶ (x0 = x1 ⟶ x2) ⟶ (x1 ∈ x0 ⟶ x2) ⟶ x2
Known ordsuccI2^ordsuccI2 : ∀ x0 . x0 ∈ ordsucc x0
Known ordsuccI1^ordsuccI1 : ∀ x0 . x0 ⊆ ordsucc x0
Theorem infinite_bigger^{infinite_bigger} : ∀ x0 . x0 ⊆ omega ⟶ infinite x0 ⟶ ∀ x1 . x1 ∈ omega ⟶ ∀ x2 : ο . (∀ x3 . and (x3 ∈ x0) (x1 ∈ x3) ⟶ x2) ⟶ x2 (proof)