Proofgold Address

Param ordinal^ordinal : ι → ο
Param mul_SNo^mul_SNo : ι → ι → ι
Param SNo^SNo : ι → ο
Param SNoS_^SNoS_ : ι → ι
Param SNoLev^SNoLev : ι → ι
Param SNoLt^SNoLt : ι → ι → ο
Known SNo_max_ordinal^{SNo_max_ordinal} : ∀ x0 . SNo x0 ⟶ (∀ x1 . x1 ∈ SNoS_ (SNoLev x0) ⟶ SNoLt x1 x0) ⟶ ordinal x0
Param SNo_^SNo_ : ι → ι → ο
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 SNoLev_ordinal^{SNoLev_ordinal} : ∀ x0 . SNo x0 ⟶ ordinal (SNoLev x0)
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
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 In_irref^In_irref : ∀ x0 . nIn x0 x0
Param SNoR^SNoR : ι → ι
Param SNoL^SNoL : ι → ι
Param SNoLe^SNoLe : ι → ι → ο
Param add_SNo^add_SNo : ι → ι → ι
Known mul_SNo_SNoR_interpolate_impred^{mul_SNo_SNoR_interpolate_impred} : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ ∀ x2 . x2 ∈ SNoR (mul_SNo x0 x1) ⟶ ∀ x3 : ο . (∀ x4 . x4 ∈ SNoL x0 ⟶ ∀ x5 . x5 ∈ SNoR x1 ⟶ SNoLe (add_SNo (mul_SNo x4 x1) (mul_SNo x0 x5)) (add_SNo x2 (mul_SNo x4 x5)) ⟶ x3) ⟶ (∀ x4 . x4 ∈ SNoR x0 ⟶ ∀ x5 . x5 ∈ SNoL x1 ⟶ SNoLe (add_SNo (mul_SNo x4 x1) (mul_SNo x0 x5)) (add_SNo x2 (mul_SNo x4 x5)) ⟶ x3) ⟶ x3
Known SNoR_I^SNoR_I : ∀ x0 . SNo x0 ⟶ ∀ x1 . SNo x1 ⟶ SNoLev x1 ∈ SNoLev x0 ⟶ SNoLt x0 x1 ⟶ x1 ∈ SNoR x0
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 ordinal_SNoLev^{ordinal_SNoLev} : ∀ x0 . ordinal x0 ⟶ SNoLev x0 = x0
Known SNoLt_irref^SNoLt_irref : ∀ x0 . not (SNoLt x0 x0)
Known SNoLt_tra^SNoLt_tra : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNoLt x0 x1 ⟶ SNoLt x1 x2 ⟶ SNoLt x0 x2
Known ordinal_SNoLev_max^{ordinal_SNoLev_max} : ∀ x0 . ordinal x0 ⟶ ∀ x1 . SNo x1 ⟶ SNoLev x1 ∈ x0 ⟶ SNoLt x1 x0
Known SNo_mul_SNo^SNo_mul_SNo : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNo (mul_SNo x0 x1)
Known ordinal_SNo^ordinal_SNo : ∀ x0 . ordinal x0 ⟶ SNo x0
Theorem mul_SNo_ordinal_ordinal^{mul_SNo_ordinal_ordinal} : ∀ x0 . ordinal x0 ⟶ ∀ x1 . ordinal x1 ⟶ ordinal (mul_SNo x0 x1) (proof)
Known ordinal_SNoLt_In^{ordinal_SNoLt_In} : ∀ x0 x1 . ordinal x0 ⟶ ordinal x1 ⟶ SNoLt x0 x1 ⟶ x0 ∈ x1
Known add_SNo_ordinal_ordinal^{add_SNo_ordinal_ordinal} : ∀ x0 . ordinal x0 ⟶ ∀ x1 . ordinal x1 ⟶ ordinal (add_SNo x0 x1)
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))
Known ordinal_In_SNoLt^{ordinal_In_SNoLt} : ∀ x0 . ordinal x0 ⟶ ∀ x1 . x1 ∈ x0 ⟶ SNoLt x1 x0
Known ordinal_Hered^{ordinal_Hered} : ∀ x0 . ordinal x0 ⟶ ∀ x1 . x1 ∈ x0 ⟶ ordinal x1
Theorem 59192.. : ∀ x0 . ordinal x0 ⟶ ∀ x1 . ordinal x1 ⟶ ∀ x2 . x2 ∈ x0 ⟶ ∀ x3 . x3 ∈ x1 ⟶ add_SNo (mul_SNo x2 x1) (mul_SNo x0 x3) ∈ add_SNo (mul_SNo x0 x1) (mul_SNo x2 x3) (proof)
Definition and^and := λ x0 x1 : ο . ∀ x2 : ο . (x0 ⟶ x1 ⟶ x2) ⟶ x2
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 Sing^Sing : ι → ι
Known and3I^and3I : ∀ x0 x1 x2 : ο . x0 ⟶ x1 ⟶ x2 ⟶ and (and x0 x1) x2
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 SingE^SingE : ∀ x0 x1 . x1 ∈ Sing x0 ⟶ x1 = x0
Theorem 10cf1.. : ∀ x0 . SNo x0 ⟶ SNoCutP (SNoL x0) (Sing x0) (proof)
Param SNo_extend0^SNo_extend0 : ι → ι
Param SNoCut^SNoCut : ι → ι → ι
Param ordsucc^ordsucc : ι → ι
Param binunion^binunion : ι → ι → ι
Param famunion^famunion : ι → (ι → ι) → ι
Definition Subq^Subq := λ x0 x1 . ∀ x2 . x2 ∈ x0 ⟶ x2 ∈ x1
Param SNoEq_^SNoEq_ : ι → ι → ι → ο
Known SNoCutP_SNoCut_impred^{SNoCutP_SNoCut_impred} : ∀ x0 x1 . SNoCutP x0 x1 ⟶ ∀ x2 : ο . (SNo (SNoCut x0 x1) ⟶ SNoLev (SNoCut x0 x1) ∈ ordsucc (binunion (famunion x0 (λ x3 . ordsucc (SNoLev x3))) (famunion x1 (λ x3 . ordsucc (SNoLev x3)))) ⟶ (∀ x3 . x3 ∈ x0 ⟶ SNoLt x3 (SNoCut x0 x1)) ⟶ (∀ x3 . x3 ∈ x1 ⟶ SNoLt (SNoCut x0 x1) x3) ⟶ (∀ x3 . SNo x3 ⟶ (∀ x4 . x4 ∈ x0 ⟶ SNoLt x4 x3) ⟶ (∀ x4 . x4 ∈ x1 ⟶ SNoLt x3 x4) ⟶ and (SNoLev (SNoCut x0 x1) ⊆ SNoLev x3) (SNoEq_ (SNoLev (SNoCut x0 x1)) (SNoCut x0 x1) x3)) ⟶ x2) ⟶ x2
Known SNo_extend0_SNo^{SNo_extend0_SNo} : ∀ x0 . SNo x0 ⟶ SNo (SNo_extend0 x0)
Param binintersect^binintersect : ι → ι → ι
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 SNoLtI2^SNoLtI2 : ∀ x0 x1 . SNoLev x0 ∈ SNoLev x1 ⟶ SNoEq_ (SNoLev x0) x0 x1 ⟶ SNoLev x0 ∈ x1 ⟶ SNoLt x0 x1
Known SNo_extend0_SNoLev^{SNo_extend0_SNoLev} : ∀ x0 . SNo x0 ⟶ SNoLev (SNo_extend0 x0) = ordsucc (SNoLev x0)
Known ordsuccI1^ordsuccI1 : ∀ x0 . x0 ⊆ ordsucc x0
Known SNoEq_tra_^SNoEq_tra_ : ∀ x0 x1 x2 x3 . SNoEq_ x0 x1 x2 ⟶ SNoEq_ x0 x2 x3 ⟶ SNoEq_ x0 x1 x3
Known SNoEq_antimon_^{SNoEq_antimon_} : ∀ x0 . ordinal x0 ⟶ ∀ x1 . x1 ∈ x0 ⟶ ∀ x2 x3 . SNoEq_ x0 x2 x3 ⟶ SNoEq_ x1 x2 x3
Known SNoEq_sym_^SNoEq_sym_ : ∀ x0 x1 x2 . SNoEq_ x0 x1 x2 ⟶ SNoEq_ x0 x2 x1
Known SNo_extend0_SNoEq^{SNo_extend0_SNoEq} : ∀ x0 . SNo x0 ⟶ SNoEq_ (SNoLev x0) (SNo_extend0 x0) x0
Known SNoEq_E2^SNoEq_E2 : ∀ x0 x1 x2 . SNoEq_ x0 x1 x2 ⟶ ∀ x3 . x3 ∈ x0 ⟶ x3 ∈ x2 ⟶ x3 ∈ x1
Known binintersectE2^{binintersectE2} : ∀ x0 x1 x2 . x2 ∈ binintersect x0 x1 ⟶ x2 ∈ x1
Known In_no2cycle^In_no2cycle : ∀ x0 x1 . x0 ∈ x1 ⟶ x1 ∈ x0 ⟶ False
Known SNo_extend0_Lt^{SNo_extend0_Lt} : ∀ x0 . SNo x0 ⟶ SNoLt (SNo_extend0 x0) x0
Known SingI^SingI : ∀ x0 . x0 ∈ Sing x0
Known binintersectE1^{binintersectE1} : ∀ x0 x1 x2 . x2 ∈ binintersect x0 x1 ⟶ x2 ∈ x0
Known SNoL_I^SNoL_I : ∀ x0 . SNo x0 ⟶ ∀ x1 . SNo x1 ⟶ SNoLev x1 ∈ SNoLev x0 ⟶ SNoLt x1 x0 ⟶ x1 ∈ SNoL x0
Known SNo_eq^SNo_eq : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNoLev x0 = SNoLev x1 ⟶ SNoEq_ (SNoLev x0) x0 x1 ⟶ x0 = x1
Known set_ext^set_ext : ∀ x0 x1 . x0 ⊆ x1 ⟶ x1 ⊆ x0 ⟶ x0 = x1
Definition or^or := λ x0 x1 : ο . ∀ x2 : ο . (x0 ⟶ x2) ⟶ (x1 ⟶ x2) ⟶ x2
Known ordsuccE^ordsuccE : ∀ x0 x1 . x1 ∈ ordsucc x0 ⟶ or (x1 ∈ x0) (x1 = x0)
Definition TransSet^TransSet := λ x0 . ∀ x1 . x1 ∈ x0 ⟶ x1 ⊆ x0
Known ordinal_TransSet^{ordinal_TransSet} : ∀ x0 . ordinal x0 ⟶ TransSet x0
Theorem 73c53.. : ∀ x0 . SNo x0 ⟶ SNo_extend0 x0 = SNoCut (SNoL x0) (Sing x0) (proof)
Theorem 8d610.. : ∀ x0 . SNo x0 ⟶ SNoCutP (Sing x0) (SNoR x0) (proof)
Param SNo_extend1^SNo_extend1 : ι → ι
Known SNo_extend1_SNo^{SNo_extend1_SNo} : ∀ x0 . SNo x0 ⟶ SNo (SNo_extend1 x0)
Known SNo_extend1_Gt^{SNo_extend1_Gt} : ∀ x0 . SNo x0 ⟶ SNoLt x0 (SNo_extend1 x0)
Known SNoLtI3^SNoLtI3 : ∀ x0 x1 . SNoLev x1 ∈ SNoLev x0 ⟶ SNoEq_ (SNoLev x1) x0 x1 ⟶ nIn (SNoLev x1) x0 ⟶ SNoLt x0 x1
Known SNo_extend1_SNoLev^{SNo_extend1_SNoLev} : ∀ x0 . SNo x0 ⟶ SNoLev (SNo_extend1 x0) = ordsucc (SNoLev x0)
Known SNo_extend1_SNoEq^{SNo_extend1_SNoEq} : ∀ x0 . SNo x0 ⟶ SNoEq_ (SNoLev x0) (SNo_extend1 x0) x0
Known SNoEq_E1^SNoEq_E1 : ∀ x0 x1 x2 . SNoEq_ x0 x1 x2 ⟶ ∀ x3 . x3 ∈ x0 ⟶ x3 ∈ x1 ⟶ x3 ∈ x2
Theorem 62b20.. : ∀ x0 . SNo x0 ⟶ SNo_extend1 x0 = SNoCut (Sing x0) (SNoR x0) (proof)