Let x0 of type ι be given.
Apply SNoLev_ordinal with
x0.
The subproof is completed by applying H0.
Apply PNoEq_tra_ with
SNoLev x0,
λ x1 . x1 ∈ PSNo (ordsucc (SNoLev x0)) (λ x2 . or (x2 ∈ x0) (x2 = SNoLev x0)),
λ x1 . or (x1 ∈ x0) (x1 = SNoLev x0),
λ x1 . x1 ∈ x0 leaving 2 subgoals.
Apply PNoEq_antimon_ with
λ x1 . x1 ∈ PSNo (ordsucc (SNoLev x0)) (λ x2 . or (x2 ∈ x0) (x2 = SNoLev x0)),
λ x1 . or (x1 ∈ x0) (x1 = SNoLev x0),
ordsucc (SNoLev x0),
SNoLev x0 leaving 3 subgoals.
Apply ordinal_ordsucc with
SNoLev x0.
The subproof is completed by applying L1.
The subproof is completed by applying ordsuccI2 with
SNoLev x0.
Apply PNoEq_PSNo with
ordsucc (SNoLev x0),
λ x1 . or (x1 ∈ x0) (x1 = SNoLev x0).
Apply ordinal_ordsucc with
SNoLev x0.
The subproof is completed by applying L1.
Apply PNoEq_sym_ with
SNoLev x0,
λ x1 . x1 ∈ x0,
λ x1 . or (x1 ∈ x0) (x1 = SNoLev x0).
The subproof is completed by applying PNo_extend1_eq with
SNoLev x0,
λ x1 . x1 ∈ x0.