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Proofgold Proof

pf
Let x0 of type ι be given.
Assume H0: SNo x0.
Let x1 of type ι be given.
Assume H1: x1 ∈ SNoR x0.
Let x2 of type ο be given.
Assume H2: SNo x1 ⟶ SNoLev x1 ∈ SNoLev x0 ⟶ SNoLt x0 x1 ⟶ x2.
Apply SepE with SNoS_ (SNoLev x0), λ x3 . SNoLt x0 x3, x1, x2 leaving 2 subgoals.
The subproof is completed by applying H1.
Assume H3: x1 ∈ SNoS_ (SNoLev x0).
Assume H4: SNoLt x0 x1.
Apply SNoS_E2 with SNoLev x0, x1, x2 leaving 3 subgoals.
Apply SNoLev_ordinal with x0.
The subproof is completed by applying H0.
The subproof is completed by applying H3.
Assume H5: SNoLev x1 ∈ SNoLev x0.
Assume H6: ordinal (SNoLev x1).
Assume H7: SNo x1.
Assume H8: SNo_ (SNoLev x1) x1.
Apply H2 leaving 3 subgoals.
The subproof is completed by applying H7.
The subproof is completed by applying H5.
The subproof is completed by applying H4.
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