Let x0 of type ι be given.
Let x1 of type ι be given.
Assume H1: x1 ∈ x0.
Apply H0 with
and (TransSet x1) (∀ x2 . x2 ∈ x1 ⟶ TransSet x2).
Assume H3:
∀ x2 . x2 ∈ x0 ⟶ TransSet x2.
Apply andI with
TransSet x1,
∀ x2 . x2 ∈ x1 ⟶ TransSet x2 leaving 2 subgoals.
Apply H3 with
x1.
The subproof is completed by applying H1.
Let x2 of type ι be given.
Assume H4: x2 ∈ x1.
Claim L5: x2 ∈ x0
Apply H2 with
x1,
x2 leaving 2 subgoals.
The subproof is completed by applying H1.
The subproof is completed by applying H4.
Apply H3 with
x2.
The subproof is completed by applying L5.