Let x0 of type ι be given.
Let x1 of type ι → ι be given.
Let x2 of type ι be given.
Assume H0:
x2 ∈ Pi x0 (λ x3 . x1 x3).
Let x3 of type ι be given.
Assume H1:
x3 ∈ Pi x0 (λ x4 . x1 x4).
Assume H2:
∀ x4 . x4 ∈ x0 ⟶ ap x2 x4 = ap x3 x4.
Apply set_ext with
x2,
x3 leaving 2 subgoals.
Apply Pi_ext_Subq with
x0,
x1,
x2,
x3 leaving 3 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
Let x4 of type ι be given.
Assume H3: x4 ∈ x0.
Apply H2 with
x4,
λ x5 x6 . x6 ⊆ ap x3 x4 leaving 2 subgoals.
The subproof is completed by applying H3.
The subproof is completed by applying Subq_ref with
ap x3 x4.
Apply Pi_ext_Subq with
x0,
x1,
x3,
x2 leaving 3 subgoals.
The subproof is completed by applying H1.
The subproof is completed by applying H0.
Let x4 of type ι be given.
Assume H3: x4 ∈ x0.
Apply H2 with
x4,
λ x5 x6 . ap x3 x4 ⊆ x6 leaving 2 subgoals.
The subproof is completed by applying H3.
The subproof is completed by applying Subq_ref with
ap x3 x4.