Proofgold Signed Transaction

Param binunion^binunion : ι → ι → ι
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)
Theorem binunionE'^binunionE : ∀ x0 x1 x2 . ∀ x3 : ο . (x2 ∈ x0 ⟶ x3) ⟶ (x2 ∈ x1 ⟶ x3) ⟶ x2 ∈ binunion x0 x1 ⟶ x3 (proof)
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 ReplE'^ReplE : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 : ι → ο . (∀ x3 . x3 ∈ x0 ⟶ x2 (x1 x3)) ⟶ ∀ x3 . x3 ∈ prim5 x0 x1 ⟶ x2 x3 (proof)
Theorem 04306.. : ∀ x0 x1 . ∀ x2 x3 : ι → ι . ∀ x4 : ι → ο . (∀ x5 . x5 ∈ x0 ⟶ x4 (x2 x5)) ⟶ (∀ x5 . x5 ∈ x1 ⟶ x4 (x3 x5)) ⟶ ∀ x5 . x5 ∈ binunion (prim5 x0 x2) (prim5 x1 x3) ⟶ x4 x5 (proof)
Param SNo^SNo : ι → ο
Param SNoLev^SNoLev : ι → ι
Param ordsucc^ordsucc : ι → ι
Param SNoLt^SNoLt : ι → ι → ο
Param SNo_extend0^SNo_extend0 : ι → ι
Param SNoLe^SNoLe : ι → ι → ο
Known SNoLtLe_or^SNoLtLe_or : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ or (SNoLt x0 x1) (SNoLe x1 x0)
Definition False^False := ∀ x0 : ο . x0
Known FalseE^FalseE : False ⟶ ∀ x0 : ο . x0
Param binintersect^binintersect : ι → ι → ι
Param SNoEq_^SNoEq_ : ι → ι → ι → ο
Definition not^not := λ x0 : ο . x0 ⟶ False
Definition nIn^nIn := λ x0 x1 . not (x0 ∈ x1)
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 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 SNo_extend0_SNo^{SNo_extend0_SNo} : ∀ x0 . SNo x0 ⟶ SNo (SNo_extend0 x0)
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)
Definition Subq^Subq := λ x0 x1 . ∀ x2 . x2 ∈ x0 ⟶ x2 ∈ x1
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
Param ordinal^ordinal : ι → ο
Known SNoEq_antimon_^{SNoEq_antimon_} : ∀ x0 . ordinal x0 ⟶ ∀ x1 . x1 ∈ x0 ⟶ ∀ x2 x3 . SNoEq_ x0 x2 x3 ⟶ SNoEq_ x1 x2 x3
Known SNoLev_ordinal^{SNoLev_ordinal} : ∀ x0 . SNo x0 ⟶ ordinal (SNoLev x0)
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 ordsuccE^ordsuccE : ∀ x0 x1 . x1 ∈ ordsucc x0 ⟶ or (x1 ∈ x0) (x1 = x0)
Known In_no2cycle^In_no2cycle : ∀ x0 x1 . x0 ∈ x1 ⟶ x1 ∈ x0 ⟶ False
Known In_irref^In_irref : ∀ x0 . nIn x0 x0
Theorem f7eb7.. : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNoLev x1 ∈ ordsucc (SNoLev x0) ⟶ SNoLt (SNo_extend0 x0) x1 ⟶ SNoLe x0 x1 (proof)
Param SNo_extend1^SNo_extend1 : ι → ι
Known SNo_extend1_SNo^{SNo_extend1_SNo} : ∀ x0 . SNo x0 ⟶ SNo (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
Known binintersectE1^{binintersectE1} : ∀ x0 x1 x2 . x2 ∈ binintersect x0 x1 ⟶ x2 ∈ x0
Theorem 3f795.. : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNoLev x1 ∈ ordsucc (SNoLev x0) ⟶ SNoLt x1 (SNo_extend1 x0) ⟶ SNoLe x1 x0 (proof)