Let x0 of type ι be given.
Apply H0 with
x0 = 0.
Let x1 of type ι → ι be given.
Apply H1 with
x0 = 0.
Assume H2: ∀ x2 . x2 ∈ x0 ⟶ x1 x2 ∈ 0.
Assume H3: ∀ x2 . x2 ∈ x0 ⟶ ∀ x3 . x3 ∈ x0 ⟶ x1 x2 = x1 x3 ⟶ x2 = x3.
Apply Empty_eq with
x0.
Let x2 of type ι be given.
Assume H4: x2 ∈ x0.
Apply EmptyE with
x1 x2.
Apply H2 with
x2.
The subproof is completed by applying H4.