Proofgold Asset

Param SNo^SNo : ι → ο
Param SNoLev^SNoLev : ι → ι
Definition and^and := λ x0 x1 : ο . ∀ x2 : ο . (x0 ⟶ x1 ⟶ x2) ⟶ x2
Definition iff^iff := λ x0 x1 : ο . and (x0 ⟶ x1) (x1 ⟶ x0)
Definition PNoEq_^PNoEq_ := λ x0 . λ x1 x2 : ι → ο . ∀ x3 . x3 ∈ x0 ⟶ iff (x1 x3) (x2 x3)
Definition SNoEq_^SNoEq_ := λ x0 x1 x2 . PNoEq_ x0 (λ x3 . x3 ∈ x1) (λ x3 . x3 ∈ x2)
Known SNo_eq^SNo_eq : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNoLev x0 = SNoLev x1 ⟶ SNoEq_ (SNoLev x0) x0 x1 ⟶ x0 = x1
Known SNo_0^SNo_0 : SNo 0
Known SNoLev_0^SNoLev_0 : SNoLev 0 = 0
Definition False^False := ∀ x0 : ο . x0
Known FalseE^FalseE : False ⟶ ∀ x0 : ο . x0
Definition not^not := λ x0 : ο . x0 ⟶ False
Definition nIn^nIn := λ x0 x1 . not (x0 ∈ x1)
Known EmptyE^EmptyE : ∀ x0 . nIn x0 0
Theorem SNoLev_0_eq_0^{SNoLev_0_eq_0} : ∀ x0 . SNo x0 ⟶ SNoLev x0 = 0 ⟶ x0 = 0 (proof)
Definition Subq^Subq := λ x0 x1 . ∀ x2 . x2 ∈ x0 ⟶ x2 ∈ x1
Param binunion^binunion : ι → ι → ι
Param Sing^Sing : ι → ι
Definition SetAdjoin^SetAdjoin := λ x0 x1 . binunion x0 (Sing x1)
Param ordsucc^ordsucc : 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))
Param binintersect^binintersect : ι → ι → ι
Known andI^andI : ∀ x0 x1 : ο . x0 ⟶ x1 ⟶ and x0 x1
Known binintersect_Subq_2^{binintersect_Subq_2} : ∀ x0 x1 . binintersect x0 x1 ⊆ x1
Known exactly1of2_E^{exactly1of2_E} : ∀ x0 x1 : ο . exactly1of2 x0 x1 ⟶ ∀ x2 : ο . (x0 ⟶ not x1 ⟶ x2) ⟶ (not x0 ⟶ x1 ⟶ x2) ⟶ x2
Known exactly1of2_I1^{exactly1of2_I1} : ∀ x0 x1 : ο . x0 ⟶ not x1 ⟶ exactly1of2 x0 x1
Known binintersectI^{binintersectI} : ∀ x0 x1 x2 . x2 ∈ x0 ⟶ x2 ∈ x1 ⟶ x2 ∈ binintersect x0 x1
Known binunionI2^binunionI2 : ∀ x0 x1 x2 . x2 ∈ x1 ⟶ x2 ∈ binunion x0 x1
Known ReplI^ReplI : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 . x2 ∈ x0 ⟶ x1 x2 ∈ prim5 x0 x1
Known binintersectE1^{binintersectE1} : ∀ x0 x1 x2 . x2 ∈ binintersect x0 x1 ⟶ x2 ∈ x0
Known exactly1of2_I2^{exactly1of2_I2} : ∀ x0 x1 : ο . not x0 ⟶ x1 ⟶ exactly1of2 x0 x1
Known binunionI1^binunionI1 : ∀ x0 x1 x2 . x2 ∈ x0 ⟶ x2 ∈ binunion x0 x1
Param ordinal^ordinal : ι → ο
Definition TransSet^TransSet := λ x0 . ∀ x1 . x1 ∈ x0 ⟶ x1 ⊆ x0
Known ordinal_TransSet^{ordinal_TransSet} : ∀ x0 . ordinal x0 ⟶ TransSet x0
Known SNoLev_ordinal^{SNoLev_ordinal} : ∀ x0 . SNo x0 ⟶ ordinal (SNoLev x0)
Known SNoLev_^SNoLev_ : ∀ x0 . SNo x0 ⟶ SNo_ (SNoLev x0) x0
Theorem restr_SNo_^restr_SNo_ : ∀ x0 . SNo x0 ⟶ ∀ x1 . x1 ∈ SNoLev x0 ⟶ SNo_ x1 (binintersect x0 (SNoElts_ x1)) (proof)
Known SNo_SNo^SNo_SNo : ∀ x0 . ordinal x0 ⟶ ∀ x1 . SNo_ x0 x1 ⟶ SNo x1
Known ordinal_Hered^{ordinal_Hered} : ∀ x0 . ordinal x0 ⟶ ∀ x1 . x1 ∈ x0 ⟶ ordinal x1
Theorem restr_SNo^restr_SNo : ∀ x0 . SNo x0 ⟶ ∀ x1 . x1 ∈ SNoLev x0 ⟶ SNo (binintersect x0 (SNoElts_ x1)) (proof)
Known SNoLev_uniq2^SNoLev_uniq2 : ∀ x0 . ordinal x0 ⟶ ∀ x1 . SNo_ x0 x1 ⟶ SNoLev x1 = x0
Theorem restr_SNoLev^restr_SNoLev : ∀ x0 . SNo x0 ⟶ ∀ x1 . x1 ∈ SNoLev x0 ⟶ SNoLev (binintersect x0 (SNoElts_ x1)) = x1 (proof)
Known SNoEq_I^SNoEq_I : ∀ x0 x1 x2 . (∀ x3 . x3 ∈ x0 ⟶ iff (x3 ∈ x1) (x3 ∈ x2)) ⟶ SNoEq_ x0 x1 x2
Known iffI^iffI : ∀ x0 x1 : ο . (x0 ⟶ x1) ⟶ (x1 ⟶ x0) ⟶ iff x0 x1
Theorem restr_SNoEq^restr_SNoEq : ∀ x0 . SNo x0 ⟶ ∀ x1 . x1 ∈ SNoLev x0 ⟶ SNoEq_ x1 (binintersect x0 (SNoElts_ x1)) x0 (proof)
Param SNoLt^SNoLt : ι → ι → ο
Definition SNoCutP^SNoCutP := λ x0 x1 . and (and (∀ x2 . x2 ∈ x0 ⟶ SNo x2) (∀ x2 . x2 ∈ x1 ⟶ SNo x2)) (∀ x2 . x2 ∈ x0 ⟶ ∀ x3 . x3 ∈ x1 ⟶ SNoLt x2 x3)
Param Sep^Sep : ι → (ι → ο) → ι
Param SNoL^SNoL : ι → ι
Param SNoR^SNoR : ι → ι
Param SNoCut^SNoCut : ι → ι → ι
Known SNo_eta^SNo_eta : ∀ x0 . SNo x0 ⟶ x0 = SNoCut (SNoL x0) (SNoR x0)
Known set_ext^set_ext : ∀ x0 x1 . x0 ⊆ x1 ⟶ x1 ⊆ x0 ⟶ x0 = x1
Known SNoL_E^SNoL_E : ∀ x0 . SNo x0 ⟶ ∀ x1 . x1 ∈ SNoL x0 ⟶ ∀ x2 : ο . (SNo x1 ⟶ SNoLev x1 ∈ SNoLev x0 ⟶ SNoLt x1 x0 ⟶ x2) ⟶ x2
Known SepI^SepI : ∀ x0 . ∀ x1 : ι → ο . ∀ x2 . x2 ∈ x0 ⟶ x1 x2 ⟶ x2 ∈ Sep x0 x1
Known SNoL_I^SNoL_I : ∀ x0 . SNo x0 ⟶ ∀ x1 . SNo x1 ⟶ SNoLev x1 ∈ SNoLev x0 ⟶ SNoLt x1 x0 ⟶ x1 ∈ SNoL x0
Known SNoLtE^SNoLtE : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNoLt x0 x1 ⟶ ∀ x2 : ο . (∀ x3 . SNo x3 ⟶ SNoLev x3 ∈ binintersect (SNoLev x0) (SNoLev x1) ⟶ SNoEq_ (SNoLev x3) x3 x0 ⟶ SNoEq_ (SNoLev x3) x3 x1 ⟶ SNoLt x0 x3 ⟶ SNoLt x3 x1 ⟶ nIn (SNoLev x3) x0 ⟶ SNoLev x3 ∈ x1 ⟶ x2) ⟶ (SNoLev x0 ∈ SNoLev x1 ⟶ SNoEq_ (SNoLev x0) x0 x1 ⟶ SNoLev x0 ∈ x1 ⟶ x2) ⟶ (SNoLev x1 ∈ SNoLev x0 ⟶ SNoEq_ (SNoLev x1) x0 x1 ⟶ nIn (SNoLev x1) x0 ⟶ x2) ⟶ x2
Known binintersectE^{binintersectE} : ∀ x0 x1 x2 . x2 ∈ binintersect x0 x1 ⟶ and (x2 ∈ x0) (x2 ∈ x1)
Known SNoLt_tra^SNoLt_tra : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNoLt x0 x1 ⟶ SNoLt x1 x2 ⟶ SNoLt x0 x2
Known SNoLtI2^SNoLtI2 : ∀ x0 x1 . SNoLev x0 ∈ SNoLev x1 ⟶ SNoEq_ (SNoLev x0) x0 x1 ⟶ SNoLev x0 ∈ x1 ⟶ SNoLt x0 x1
Known In_no2cycle^In_no2cycle : ∀ x0 x1 . x0 ∈ x1 ⟶ x1 ∈ x0 ⟶ False
Known SepE^SepE : ∀ x0 . ∀ x1 : ι → ο . ∀ x2 . x2 ∈ Sep x0 x1 ⟶ and (x2 ∈ x0) (x1 x2)
Known SNoR_E^SNoR_E : ∀ x0 . SNo x0 ⟶ ∀ x1 . x1 ∈ SNoR x0 ⟶ ∀ x2 : ο . (SNo x1 ⟶ SNoLev x1 ∈ SNoLev x0 ⟶ SNoLt x0 x1 ⟶ x2) ⟶ x2
Known SNoR_I^SNoR_I : ∀ x0 . SNo x0 ⟶ ∀ x1 . SNo x1 ⟶ SNoLev x1 ∈ SNoLev x0 ⟶ SNoLt x0 x1 ⟶ x1 ∈ SNoR x0
Known SNoLtI3^SNoLtI3 : ∀ x0 x1 . SNoLev x1 ∈ SNoLev x0 ⟶ SNoEq_ (SNoLev x1) x0 x1 ⟶ nIn (SNoLev x1) x0 ⟶ SNoLt x0 x1
Known and3I^and3I : ∀ x0 x1 x2 : ο . x0 ⟶ x1 ⟶ x2 ⟶ and (and x0 x1) x2
Theorem restr_SNo_SNoCut^{restr_SNo_SNoCut} : ∀ x0 . SNo x0 ⟶ ∀ x1 . x1 ∈ SNoLev x0 ⟶ ∀ x2 : ο . (SNoCutP {x3 ∈ SNoL x0|SNoLev x3 ∈ x1} {x3 ∈ SNoR x0|SNoLev x3 ∈ x1} ⟶ binintersect x0 (SNoElts_ x1) = SNoCut {x4 ∈ SNoL x0|SNoLev x4 ∈ x1} {x4 ∈ SNoR x0|SNoLev x4 ∈ x1} ⟶ x2) ⟶ x2 (proof)
Param SNoS_^SNoS_ : ι → ι
Param omega^omega : ι
Param minus_SNo^minus_SNo : ι → ι
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 omega_ordinal^{omega_ordinal} : ordinal omega
Known SNoS_I^SNoS_I : ∀ x0 . ordinal x0 ⟶ ∀ x1 x2 . x2 ∈ x0 ⟶ SNo_ x2 x1 ⟶ x1 ∈ SNoS_ x0
Known minus_SNo_Lev^{minus_SNo_Lev} : ∀ x0 . SNo x0 ⟶ SNoLev (minus_SNo x0) = SNoLev x0
Known SNo_minus_SNo^{SNo_minus_SNo} : ∀ x0 . SNo x0 ⟶ SNo (minus_SNo x0)
Theorem minus_SNo_SNoS_omega^{minus_SNo_SNoS_omega} : ∀ x0 . x0 ∈ SNoS_ omega ⟶ minus_SNo x0 ∈ SNoS_ omega (proof)
Definition eps_^eps_ := λ x0 . binunion (Sing 0) {SetAdjoin (ordsucc x1) (Sing 1)|x1 ∈ x0}
Definition or^or := λ x0 x1 : ο . ∀ x2 : ο . (x0 ⟶ x2) ⟶ (x1 ⟶ x2) ⟶ x2
Known binunionE^binunionE : ∀ x0 x1 x2 . x2 ∈ binunion x0 x1 ⟶ or (x2 ∈ x0) (x2 ∈ x1)
Known SingE^SingE : ∀ x0 x1 . x1 ∈ Sing x0 ⟶ x1 = 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 tagged_not_ordinal^{tagged_not_ordinal} : ∀ x0 . not (ordinal (SetAdjoin x0 (Sing 1)))
Theorem eps_ordinal_In_eq_0^{eps_ordinal_In_eq_0} : ∀ x0 x1 . ordinal x1 ⟶ x1 ∈ eps_ x0 ⟶ x1 = 0 (proof)
Known In_0_1^In_0_1 : 0 ∈ 1
Known cases_1^cases_1 : ∀ x0 . x0 ∈ 1 ⟶ ∀ x1 : ι → ο . x1 0 ⟶ x1 x0
Known SingI^SingI : ∀ x0 . x0 ∈ Sing x0
Theorem eps_0_1^eps_0_1 : eps_ 0 = 1 (proof)
Param nat_p^nat_p : ι → ο
Known nat_0_in_ordsucc^{nat_0_in_ordsucc} : ∀ x0 . nat_p x0 ⟶ 0 ∈ ordsucc x0
Known nat_ordsucc_in_ordsucc^{nat_ordsucc_in_ordsucc} : ∀ x0 . nat_p x0 ⟶ ∀ x1 . x1 ∈ x0 ⟶ ordsucc x1 ∈ ordsucc x0
Known nat_inv^nat_inv : ∀ x0 . nat_p x0 ⟶ or (x0 = 0) (∀ x1 : ο . (∀ x2 . and (nat_p x2) (x0 = ordsucc x2) ⟶ x1) ⟶ x1)
Known neq_0_ordsucc^{neq_0_ordsucc} : ∀ x0 . 0 = ordsucc x0 ⟶ ∀ x1 : ο . x1
Known tagged_eqE_eq^{tagged_eqE_eq} : ∀ x0 x1 . ordinal x0 ⟶ ordinal x1 ⟶ SetAdjoin x0 (Sing 1) = SetAdjoin x1 (Sing 1) ⟶ x0 = x1
Known ordinal_Empty^{ordinal_Empty} : ordinal 0
Known nat_p_ordinal^{nat_p_ordinal} : ∀ x0 . nat_p x0 ⟶ ordinal x0
Known nat_ordsucc^nat_ordsucc : ∀ x0 . nat_p x0 ⟶ nat_p (ordsucc x0)
Known nat_p_trans^nat_p_trans : ∀ x0 . nat_p x0 ⟶ ∀ x1 . x1 ∈ x0 ⟶ nat_p x1
Known ordsuccE^ordsuccE : ∀ x0 x1 . x1 ∈ ordsucc x0 ⟶ or (x1 ∈ x0) (x1 = x0)
Known nat_trans^nat_trans : ∀ x0 . nat_p x0 ⟶ ∀ x1 . x1 ∈ x0 ⟶ x1 ⊆ x0
Known ordsuccI2^ordsuccI2 : ∀ x0 . x0 ∈ ordsucc x0
Known omega_nat_p^omega_nat_p : ∀ x0 . x0 ∈ omega ⟶ nat_p x0
Theorem SNo__eps_^SNo__eps_ : ∀ x0 . x0 ∈ omega ⟶ SNo_ (ordsucc x0) (eps_ x0) (proof)
Known ordinal_ordsucc^{ordinal_ordsucc} : ∀ x0 . ordinal x0 ⟶ ordinal (ordsucc x0)
Theorem SNo_eps_^SNo_eps_ : ∀ x0 . x0 ∈ omega ⟶ SNo (eps_ x0) (proof)
Theorem SNoLev_eps_^SNoLev_eps_ : ∀ x0 . x0 ∈ omega ⟶ SNoLev (eps_ x0) = ordsucc x0 (proof)
Known omega_ordsucc^{omega_ordsucc} : ∀ x0 . x0 ∈ omega ⟶ ordsucc x0 ∈ omega
Theorem SNo_eps_SNoS_omega^{SNo_eps_SNoS_omega} : ∀ x0 . x0 ∈ omega ⟶ eps_ x0 ∈ SNoS_ omega (proof)
Known neq_ordsucc_0^{neq_ordsucc_0} : ∀ x0 . ordsucc x0 = 0 ⟶ ∀ x1 : ο . x1
Known nat_p_omega^nat_p_omega : ∀ x0 . nat_p x0 ⟶ x0 ∈ omega
Theorem SNo_eps_decr^SNo_eps_decr : ∀ x0 . x0 ∈ omega ⟶ ∀ x1 . x1 ∈ x0 ⟶ SNoLt (eps_ x0) (eps_ x1) (proof)
Theorem SNo_eps_pos^SNo_eps_pos : ∀ x0 . x0 ∈ omega ⟶ SNoLt 0 (eps_ x0) (proof)
Known nat_complete_ind^{nat_complete_ind} : ∀ x0 : ι → ο . (∀ x1 . nat_p x1 ⟶ (∀ x2 . x2 ∈ x1 ⟶ x0 x2) ⟶ x0 x1) ⟶ ∀ x1 . nat_p x1 ⟶ x0 x1
Known SNoLt_trichotomy_or_impred^{SNoLt_trichotomy_or_impred} : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ ∀ x2 : ο . (SNoLt x0 x1 ⟶ x2) ⟶ (x0 = x1 ⟶ x2) ⟶ (SNoLt x1 x0 ⟶ x2) ⟶ x2
Known SNoLt_irref^SNoLt_irref : ∀ x0 . not (SNoLt x0 x0)
Known ordinal_ordsucc_In_Subq^{ordinal_ordsucc_In_Subq} : ∀ x0 . ordinal x0 ⟶ ∀ x1 . x1 ∈ x0 ⟶ ordsucc x1 ⊆ x0
Known In_no3cycle^In_no3cycle : ∀ x0 x1 x2 . x0 ∈ x1 ⟶ x1 ∈ x2 ⟶ x2 ∈ x0 ⟶ False
Theorem d115f..^{SNo_pos_eps_Lt} : ∀ x0 . nat_p x0 ⟶ ∀ x1 . x1 ∈ SNoS_ x0 ⟶ SNoLt 0 x1 ⟶ SNoLt (eps_ x0) x1 (proof)
Param add_SNo^add_SNo : ι → ι → ι
Known add_SNo_com^add_SNo_com : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ add_SNo x0 x1 = add_SNo x1 x0
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_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 add_SNo_0L^add_SNo_0L : ∀ x0 . SNo x0 ⟶ add_SNo 0 x0 = x0
Known add_SNo_SNoS_omega^{add_SNo_SNoS_omega} : ∀ x0 . x0 ∈ SNoS_ omega ⟶ ∀ x1 . x1 ∈ SNoS_ omega ⟶ add_SNo x0 x1 ∈ SNoS_ omega
Theorem add_SNo_omega_eps_Lt^{add_SNo_omega_eps_Lt} : ∀ x0 . x0 ∈ SNoS_ omega ⟶ ∀ x1 . x1 ∈ SNoS_ omega ⟶ SNoLt x0 x1 ⟶ ∀ x2 : ο . (∀ x3 . and (x3 ∈ omega) (SNoLt (add_SNo x0 (eps_ x3)) x1) ⟶ x2) ⟶ x2 (proof)
Definition f8473..^diadic_open := λ x0 . and (x0 ⊆ SNoS_ omega) (∀ x1 . x1 ∈ x0 ⟶ ∀ x2 : ο . (∀ x3 . and (x3 ∈ omega) (∀ x4 . x4 ∈ SNoS_ omega ⟶ SNoLt (add_SNo x1 (minus_SNo (eps_ x3))) x4 ⟶ SNoLt x4 (add_SNo x1 (eps_ x3)) ⟶ x4 ∈ x0) ⟶ x2) ⟶ x2)
Theorem c974c..^{diadic_open_I} : ∀ x0 . x0 ⊆ SNoS_ omega ⟶ (∀ x1 . x1 ∈ x0 ⟶ ∀ x2 : ο . (∀ x3 . and (x3 ∈ omega) (∀ x4 . x4 ∈ SNoS_ omega ⟶ SNoLt (add_SNo x1 (minus_SNo (eps_ x3))) x4 ⟶ SNoLt x4 (add_SNo x1 (eps_ x3)) ⟶ x4 ∈ x0) ⟶ x2) ⟶ x2) ⟶ f8473.. x0 (proof)
Definition SNoL_omega^SNoL_omega := λ x0 . {x1 ∈ SNoS_ omega|SNoLt x1 x0}
Definition SNoR_omega^SNoR_omega := λ x0 . Sep (SNoS_ omega) (SNoLt x0)
Known SepE1^SepE1 : ∀ x0 . ∀ x1 : ι → ο . ∀ x2 . x2 ∈ Sep x0 x1 ⟶ x2 ∈ x0
Known SNo_add_SNo^SNo_add_SNo : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNo (add_SNo x0 x1)
Theorem 7fb68..^{diadic_open_SNoL_omega_I} : ∀ x0 . SNo x0 ⟶ (∀ x1 . x1 ∈ SNoL_omega x0 ⟶ ∀ x2 : ο . (∀ x3 . and (x3 ∈ omega) (SNoLt (add_SNo x1 (eps_ x3)) x0) ⟶ x2) ⟶ x2) ⟶ f8473.. (SNoL_omega x0) (proof)
Theorem 58c46..^{diadic_open_SNoR_omega_I} : ∀ x0 . SNo x0 ⟶ (∀ x1 . x1 ∈ SNoR_omega x0 ⟶ ∀ x2 : ο . (∀ x3 . and (x3 ∈ omega) (SNoLt x0 (add_SNo x1 (minus_SNo (eps_ x3)))) ⟶ x2) ⟶ x2) ⟶ f8473.. (SNoR_omega x0) (proof)
Definition 87ab7..^real := {x0 ∈ SNoS_ (ordsucc omega)|and (and (and (SNoL_omega x0 = 0 ⟶ ∀ x1 : ο . x1) (SNoR_omega x0 = 0 ⟶ ∀ x1 : ο . x1)) (f8473.. (SNoL_omega x0))) (f8473.. (SNoR_omega x0))}
Known and4I^and4I : ∀ x0 x1 x2 x3 : ο . x0 ⟶ x1 ⟶ x2 ⟶ x3 ⟶ and (and (and x0 x1) x2) x3
Theorem 59424..^real_I : ∀ x0 . x0 ∈ SNoS_ (ordsucc omega) ⟶ (SNoL_omega x0 = 0 ⟶ ∀ x1 : ο . x1) ⟶ (SNoR_omega x0 = 0 ⟶ ∀ x1 : ο . x1) ⟶ f8473.. (SNoL_omega x0) ⟶ f8473.. (SNoR_omega x0) ⟶ x0 ∈ 87ab7.. (proof)
Known and4E^and4E : ∀ x0 x1 x2 x3 : ο . and (and (and x0 x1) x2) x3 ⟶ ∀ x4 : ο . (x0 ⟶ x1 ⟶ x2 ⟶ x3 ⟶ x4) ⟶ x4
Theorem e2acb..^real_E : ∀ x0 . x0 ∈ 87ab7.. ⟶ ∀ x1 : ο . (x0 ∈ SNoS_ (ordsucc omega) ⟶ (SNoL_omega x0 = 0 ⟶ ∀ x2 : ο . x2) ⟶ (SNoR_omega x0 = 0 ⟶ ∀ x2 : ο . x2) ⟶ f8473.. (SNoL_omega x0) ⟶ f8473.. (SNoR_omega x0) ⟶ x1) ⟶ x1 (proof)
Known Sep_Subq^Sep_Subq : ∀ x0 . ∀ x1 : ι → ο . Sep x0 x1 ⊆ x0
Theorem 2858a..^{Subq_real_SNoS_ordsucc_omega} : 87ab7.. ⊆ SNoS_ (ordsucc omega) (proof)
Known ordsuccI1^ordsuccI1 : ∀ x0 . x0 ⊆ ordsucc x0
Param SNo_extend0^SNo_extend0 : ι → ι
Known SNo_extend0_SNo_^{SNo_extend0_SNo_} : ∀ x0 . SNo x0 ⟶ SNo_ (ordsucc (SNoLev x0)) (SNo_extend0 x0)
Known SNo_extend0_Lt^{SNo_extend0_Lt} : ∀ x0 . SNo x0 ⟶ SNoLt (SNo_extend0 x0) x0
Param SNo_extend1^SNo_extend1 : ι → ι
Known SNo_extend1_SNo_^{SNo_extend1_SNo_} : ∀ x0 . SNo x0 ⟶ SNo_ (ordsucc (SNoLev x0)) (SNo_extend1 x0)
Known SNo_extend1_Gt^{SNo_extend1_Gt} : ∀ x0 . SNo x0 ⟶ SNoLt x0 (SNo_extend1 x0)
Theorem 73158..^{Subq_SNoS_omega_real} : ∀ x0 . x0 ∈ SNoS_ omega ⟶ x0 ∈ 87ab7.. (proof)
Theorem 354d7..^{SNoCutP_SNoL_SNoR_omega} : ∀ x0 . SNo x0 ⟶ SNoCutP (SNoL_omega x0) (SNoR_omega x0) (proof)
Known SNoS_E^SNoS_E : ∀ x0 . ordinal x0 ⟶ ∀ x1 . x1 ∈ SNoS_ x0 ⟶ ∀ x2 : ο . (∀ x3 . and (x3 ∈ x0) (SNo_ x3 x1) ⟶ x2) ⟶ x2
Known ordsucc_omega_ordinal^{ordsucc_omega_ordinal} : ordinal (ordsucc omega)
Known SNoCut_ext^SNoCut_ext : ∀ x0 x1 x2 x3 . SNoCutP x0 x1 ⟶ SNoCutP x2 x3 ⟶ (∀ x4 . x4 ∈ x0 ⟶ SNoLt x4 (SNoCut x2 x3)) ⟶ (∀ x4 . x4 ∈ x1 ⟶ SNoLt (SNoCut x2 x3) x4) ⟶ (∀ x4 . x4 ∈ x2 ⟶ SNoLt x4 (SNoCut x0 x1)) ⟶ (∀ x4 . x4 ∈ x3 ⟶ SNoLt (SNoCut x0 x1) x4) ⟶ SNoCut x0 x1 = SNoCut x2 x3
Known SNoCutP_SNoL_SNoR^{SNoCutP_SNoL_SNoR} : ∀ x0 . SNo x0 ⟶ SNoCutP (SNoL x0) (SNoR x0)
Known SNoCutP_SNoCut_L^{SNoCutP_SNoCut_L} : ∀ x0 x1 . SNoCutP x0 x1 ⟶ ∀ x2 . x2 ∈ x0 ⟶ SNoLt x2 (SNoCut x0 x1)
Known SNoCutP_SNoCut_R^{SNoCutP_SNoCut_R} : ∀ x0 x1 . SNoCutP x0 x1 ⟶ ∀ x2 . x2 ∈ x1 ⟶ SNoLt (SNoCut x0 x1) x2
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 SNoLev_uniq^SNoLev_uniq : ∀ x0 x1 x2 . ordinal x1 ⟶ ordinal x2 ⟶ SNo_ x1 x0 ⟶ SNo_ x2 x0 ⟶ x1 = x2
Theorem 1df44..^{SNoS_ordsucc_omega_SNoL_SNoR_omega} : ∀ x0 . x0 ∈ SNoS_ (ordsucc omega) ⟶ x0 = SNoCut (SNoL_omega x0) (SNoR_omega x0) (proof)
Theorem 11dba..^{real_SNoL_SNoR_omega} : ∀ x0 . x0 ∈ 87ab7.. ⟶ x0 = SNoCut (SNoL_omega x0) (SNoR_omega x0) (proof)
Known dneg^dneg : ∀ x0 : ο . not (not x0) ⟶ x0
Known Empty_eq^Empty_eq : ∀ x0 . (∀ x1 . nIn x1 x0) ⟶ x0 = 0
Theorem 95fe1..^{real_ex_diad_Lt} : ∀ x0 . x0 ∈ 87ab7.. ⟶ ∀ x1 : ο . (∀ x2 . and (x2 ∈ SNoS_ omega) (SNoLt x2 x0) ⟶ x1) ⟶ x1 (proof)
Theorem 8b827..^{real_ex_diad_Gt} : ∀ x0 . x0 ∈ 87ab7.. ⟶ ∀ x1 : ο . (∀ x2 . and (x2 ∈ SNoS_ omega) (SNoLt x0 x2) ⟶ x1) ⟶ x1 (proof)
Param mul_SNo^mul_SNo : ι → ι → ι
Known add_SNo_0R^add_SNo_0R : ∀ x0 . SNo x0 ⟶ add_SNo x0 0 = x0
Known SNo_mul_SNo^SNo_mul_SNo : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNo (mul_SNo x0 x1)
Known mul_SNo_zeroR^{mul_SNo_zeroR} : ∀ x0 . SNo x0 ⟶ mul_SNo x0 0 = 0
Known mul_SNo_com^mul_SNo_com : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ mul_SNo x0 x1 = mul_SNo x1 x0
Known mul_SNo_Lt^mul_SNo_Lt : ∀ x0 x1 x2 x3 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNo x3 ⟶ SNoLt x2 x0 ⟶ SNoLt x3 x1 ⟶ SNoLt (add_SNo (mul_SNo x2 x1) (mul_SNo x0 x3)) (add_SNo (mul_SNo x0 x1) (mul_SNo x2 x3))
Theorem mul_SNo_pos_pos^{mul_SNo_pos_pos} : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNoLt 0 x0 ⟶ SNoLt 0 x1 ⟶ SNoLt 0 (mul_SNo x0 x1) (proof)
Theorem mul_SNo_pos_neg^{mul_SNo_pos_neg} : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNoLt 0 x0 ⟶ SNoLt x1 0 ⟶ SNoLt (mul_SNo x0 x1) 0 (proof)
Theorem mul_SNo_neg_pos^{mul_SNo_neg_pos} : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNoLt x0 0 ⟶ SNoLt 0 x1 ⟶ SNoLt (mul_SNo x0 x1) 0 (proof)
Theorem mul_SNo_neg_neg^{mul_SNo_neg_neg} : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNoLt x0 0 ⟶ SNoLt x1 0 ⟶ SNoLt 0 (mul_SNo x0 x1) (proof)
Theorem mul_SNo_nonzero^{mul_SNo_nonzero} : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ (x0 = 0 ⟶ ∀ x2 : ο . x2) ⟶ (x1 = 0 ⟶ ∀ x2 : ο . x2) ⟶ mul_SNo x0 x1 = 0 ⟶ ∀ x2 : ο . x2 (proof)
Known minus_SNoCut_eq^{minus_SNoCut_eq} : ∀ x0 x1 . SNoCutP x0 x1 ⟶ minus_SNo (SNoCut x0 x1) = SNoCut (prim5 x1 minus_SNo) (prim5 x0 minus_SNo)
Known minus_SNo_invol^{minus_SNo_invol} : ∀ x0 . SNo x0 ⟶ minus_SNo (minus_SNo x0) = x0
Known minus_SNo_Lt_contra2^{minus_SNo_Lt_contra2} : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNoLt x0 (minus_SNo x1) ⟶ SNoLt x1 (minus_SNo x0)
Known minus_SNo_Lt_contra^{minus_SNo_Lt_contra} : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNoLt x0 x1 ⟶ SNoLt (minus_SNo x1) (minus_SNo x0)
Known minus_SNo_Lt_contra1^{minus_SNo_Lt_contra1} : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNoLt (minus_SNo x0) x1 ⟶ SNoLt (minus_SNo x1) x0
Theorem minus_SNo_restr_SNo^{minus_SNo_restr_SNo} : ∀ x0 . SNo x0 ⟶ ∀ x1 . x1 ∈ SNoLev x0 ⟶ binintersect (minus_SNo x0) (SNoElts_ x1) = minus_SNo (binintersect x0 (SNoElts_ x1)) (proof)
Known xm^xm : ∀ x0 : ο . or x0 (not x0)
Known SNoEq_sym_^SNoEq_sym_ : ∀ x0 x1 x2 . SNoEq_ x0 x1 x2 ⟶ SNoEq_ x0 x2 x1
Theorem minus_SNo_exactly1of2^{minus_SNo_exactly1of2} : ∀ x0 . SNo x0 ⟶ ∀ x1 . x1 ∈ SNoLev x0 ⟶ exactly1of2 (x1 ∈ x0) (x1 ∈ minus_SNo x0) (proof)
Theorem minus_SNo_In^minus_SNo_In : ∀ x0 . SNo x0 ⟶ ∀ x1 . x1 ∈ SNoLev x0 ⟶ x1 ∈ x0 ⟶ nIn x1 (minus_SNo x0) (proof)
Theorem minus_SNo_nIn^{minus_SNo_nIn} : ∀ x0 . SNo x0 ⟶ ∀ x1 . x1 ∈ SNoLev x0 ⟶ nIn x1 x0 ⟶ x1 ∈ minus_SNo x0 (proof)
Known minus_add_SNo_distr^{minus_add_SNo_distr} : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ minus_SNo (add_SNo x0 x1) = add_SNo (minus_SNo x0) (minus_SNo x1)
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 minus_SNo_0^minus_SNo_0 : minus_SNo 0 = 0
Known add_SNo_minus_Lt1b^{add_SNo_minus_Lt1b} : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNoLt x0 (add_SNo x2 x1) ⟶ SNoLt (add_SNo x0 (minus_SNo x1)) x2
Known add_SNo_minus_Lt1^{add_SNo_minus_Lt1} : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNoLt (add_SNo x0 (minus_SNo x1)) x2 ⟶ SNoLt x0 (add_SNo x2 x1)
Theorem 54413..^{real_minus_SNo} : ∀ x0 . x0 ∈ 87ab7.. ⟶ minus_SNo x0 ∈ 87ab7.. (proof)