Let x0 of type ι be given.
Let x1 of type ι be given.
Assume H2: x0 ⊆ x1.
Let x2 of type ι be given.
Assume H3:
x2 ∈ SNoS_ x0.
Apply SNoS_E with
x0,
x2,
x2 ∈ SNoS_ x1 leaving 3 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H3.
Let x3 of type ι be given.
Assume H4:
(λ x4 . and (x4 ∈ x0) (SNo_ x4 x2)) x3.
Apply H4 with
x2 ∈ SNoS_ x1.
Assume H5: x3 ∈ x0.
Apply SNoS_I with
x1,
x2,
x3 leaving 3 subgoals.
The subproof is completed by applying H1.
Apply H2 with
x3.
The subproof is completed by applying H5.
The subproof is completed by applying H6.