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.
Apply Eps_i_ex with
λ x3 . and (x1 x3) (x2 = pair_tag x0 (CD_proj0 x0 x1 x2) x3).
Apply CD_carr_E with
x0,
x1,
x2,
λ x3 . ∃ x4 . and (x1 x4) (x3 = pair_tag x0 (CD_proj0 x0 x1 x3) x4) leaving 3 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
Let x3 of type ι be given.
Let x4 of type ι be given.
Assume H2: x1 x3.
Assume H3: x1 x4.
Let x5 of type ο be given.
Apply H5 with
x4.
Apply andI with
x1 x4,
pair_tag x0 x3 x4 = pair_tag x0 (CD_proj0 x0 x1 (pair_tag x0 x3 x4)) x4 leaving 2 subgoals.
The subproof is completed by applying H3.
Apply CD_proj0_2 with
x0,
x1,
x3,
x4,
λ x6 x7 . pair_tag x0 x3 x4 = pair_tag x0 x7 x4 leaving 4 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H2.
The subproof is completed by applying H3.
Let x6 of type ι → ι → ο be given.
The subproof is completed by applying H6.