Apply ordinal_ind with
λ x0 . ∀ x1 . x1 ∈ SNoS_ x0 ⟶ SNoLev (minus_SNo x1) ⊆ SNoLev x1.
Let x0 of type ι be given.
Apply H0 with
(∀ x1 . x1 ∈ x0 ⟶ ∀ x2 . x2 ∈ SNoS_ x1 ⟶ SNoLev (minus_SNo x2) ⊆ SNoLev x2) ⟶ ∀ x1 . x1 ∈ SNoS_ x0 ⟶ SNoLev (minus_SNo x1) ⊆ SNoLev x1.
Assume H2:
∀ x1 . x1 ∈ x0 ⟶ TransSet x1.
Let x1 of type ι be given.
Assume H4:
x1 ∈ SNoS_ x0.
Apply SNoS_E2 with
x0,
x1,
SNoLev (minus_SNo x1) ⊆ SNoLev x1 leaving 3 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H4.
Apply minus_SNo_eq with
x1,
λ x2 x3 . SNoLev x3 ⊆ SNoLev x1 leaving 2 subgoals.
The subproof is completed by applying H7.