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Proofgold Proof

pf
Let x0 of type ι be given.
Let x1 of type ι be given.
Assume H0: SNoCutP x0 x1.
Apply H0 with and (and (and (and (SNo (SNoCut x0 x1)) (SNoLev (SNoCut x0 x1) ∈ ordsucc (binunion (famunion x0 (λ x2 . ordsucc (SNoLev x2))) (famunion x1 (λ x2 . ordsucc (SNoLev x2)))))) (∀ x2 . x2 ∈ x0 ⟶ SNoLt x2 (SNoCut x0 x1))) (∀ x2 . x2 ∈ x1 ⟶ SNoLt (SNoCut x0 x1) x2)) (∀ x2 . SNo x2 ⟶ (∀ x3 . x3 ∈ x0 ⟶ SNoLt x3 x2) ⟶ (∀ x3 . x3 ∈ x1 ⟶ SNoLt x2 x3) ⟶ and (SNoLev (SNoCut x0 x1) ⊆ SNoLev x2) (SNoEq_ (SNoLev (SNoCut x0 x1)) (SNoCut x0 x1) x2)).
Assume H1: and (∀ x2 . x2 ∈ x0 ⟶ SNo x2) (∀ x2 . x2 ∈ x1 ⟶ SNo x2).
Apply H1 with (∀ x2 . x2 ∈ x0 ⟶ ∀ x3 . x3 ∈ x1 ⟶ SNoLt x2 x3) ⟶ and (and (and (and (SNo (SNoCut x0 x1)) (SNoLev (SNoCut x0 x1) ∈ ordsucc (binunion (famunion x0 (λ x2 . ordsucc (SNoLev x2))) (famunion x1 (λ x2 . ordsucc (SNoLev x2)))))) (∀ x2 . x2 ∈ x0 ⟶ SNoLt x2 (SNoCut x0 x1))) (∀ x2 . x2 ∈ x1 ⟶ SNoLt (SNoCut x0 x1) x2)) (∀ x2 . SNo x2 ⟶ (∀ x3 . x3 ∈ x0 ⟶ SNoLt x3 x2) ⟶ (∀ x3 . x3 ∈ x1 ⟶ SNoLt x2 x3) ⟶ and (SNoLev (SNoCut x0 x1) ⊆ SNoLev x2) (SNoEq_ (SNoLev (SNoCut x0 x1)) (SNoCut x0 x1) x2)).
Assume H2: ∀ x2 . x2 ∈ x0 ⟶ SNo x2.
Assume H3: ∀ x2 . x2 ∈ x1 ⟶ SNo x2.
Assume H4: ∀ x2 . x2 ∈ x0 ⟶ ∀ x3 . x3 ∈ x1 ⟶ SNoLt x2 x3.
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Apply PNo_bd_pred with λ x2 . λ x3 : ι → ο . and (ordinal x2) (PSNo x2 x3 ∈ x0), λ x2 . λ x3 : ι → ο . and (ordinal x2) (PSNo x2 x3 ∈ x1), binunion (famunion x0 (λ x2 . ordsucc (SNoLev x2))) (famunion x1 (λ x2 . ordsucc (SNoLev x2))), and (and (and (and (SNo (SNoCut x0 x1)) (SNoLev (SNoCut x0 x1) ∈ ordsucc (binunion (famunion x0 (λ x2 . ordsucc ...)) ...))) ...) ...) ... leaving 5 subgoals.
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