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.
Let x4 of type ι be given.
Assume H2: x4 ∈ x2.
Let x5 of type ι be given.
Let x6 of type ι be given.
Assume H4: x6 ∈ x4.
Apply H3 with
λ x7 x8 . x6 ∈ x7.
Apply binunionI1 with
x4,
Sing x0,
x6.
The subproof is completed by applying H4.
Apply binunionE with
x5,
Sing x0,
x6,
x6 ∈ x5 leaving 3 subgoals.
The subproof is completed by applying L5.
Assume H6: x6 ∈ x5.
The subproof is completed by applying H6.
Apply FalseE with
x6 ∈ x5.
Claim L7: x6 = x0
Apply SingE with
x0,
x6.
The subproof is completed by applying H6.
Apply H0 with
x2,
x4 leaving 3 subgoals.
The subproof is completed by applying H1.
The subproof is completed by applying H2.
Apply L7 with
λ x7 x8 . x7 ∈ x4.
The subproof is completed by applying H4.