Let x0 of type ι be given.
Let x1 of type ι be given.
Assume H1:
x1 ∈ SNoS_ x0.
Let x2 of type ο be given.
Apply SNoS_E with
x0,
x1,
x2 leaving 3 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
Let x3 of type ι be given.
Assume H3:
(λ x4 . and (x4 ∈ x0) (SNo_ x4 x1)) x3.
Apply H3 with
x2.
Assume H4: x3 ∈ x0.
Apply ordinal_Hered with
x0,
x3 leaving 2 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H4.
Apply SNo_SNo with
x3,
x1 leaving 2 subgoals.
The subproof is completed by applying L6.
The subproof is completed by applying H5.
Apply SNoLev_prop with
x1,
x2 leaving 2 subgoals.
The subproof is completed by applying L7.
Apply SNoLev_uniq2 with
x3,
x1 leaving 2 subgoals.
The subproof is completed by applying L6.
The subproof is completed by applying H5.
Apply L10 with
λ x4 x5 . x5 ∈ x0.
The subproof is completed by applying H4.
Apply H2 leaving 4 subgoals.
The subproof is completed by applying L11.
The subproof is completed by applying H8.
The subproof is completed by applying L7.
The subproof is completed by applying H9.