Let x0 of type ι be given.
Let x1 of type ι → ο be given.
Assume H0:
∀ x2 . x1 x2 ⟶ ∀ x3 . x3 ∈ x2 ⟶ nIn x0 x3.
Let x2 of type ι be given.
Let x3 of type ι be given.
Assume H1: x1 x2.
Apply H0 with
x2,
(λ x4 . SetAdjoin x4 x0) x3 leaving 3 subgoals.
The subproof is completed by applying H1.
The subproof is completed by applying H2.
Apply binunionI2 with
x3,
Sing x0,
x0.
The subproof is completed by applying SingI with x0.