Let x0 of type ι be given.
Let x1 of type ι be given.
Let x2 of type ι be given.
Assume H1: x2 ∈ x0.
Apply H2 with
x1 ∈ SNoS_ x0.
Apply SepI with
prim4 (SNoElts_ x0),
λ x3 . ∃ x4 . and (x4 ∈ x0) (SNo_ x4 x3),
x1 leaving 2 subgoals.
Apply PowerI with
SNoElts_ x0,
x1.
Apply Subq_tra with
x1,
SNoElts_ x2,
SNoElts_ x0 leaving 2 subgoals.
The subproof is completed by applying H3.
Apply SNoElts_mon with
x2,
x0.
Apply H0 with
x2 ⊆ x0.
Assume H6:
∀ x3 . x3 ∈ x0 ⟶ TransSet x3.
Apply H5 with
x2.
The subproof is completed by applying H1.
Let x3 of type ο be given.
Assume H5:
∀ x4 . and (x4 ∈ x0) (SNo_ x4 x1) ⟶ x3.
Apply H5 with
x2.
Apply andI with
x2 ∈ x0,
SNo_ x2 x1 leaving 2 subgoals.
The subproof is completed by applying H1.
The subproof is completed by applying H2.