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.
Assume H2: x1 x3.
Apply CD_proj0_1 with
x0,
x1,
pair_tag x0 x2 x3,
CD_proj0 x0 x1 (pair_tag x0 x2 x3) = x2 leaving 3 subgoals.
The subproof is completed by applying H0.
Apply CD_carr_I with
x0,
x1,
x2,
x3 leaving 3 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
The subproof is completed by applying H2.
Apply H4 with
CD_proj0 x0 x1 (pair_tag x0 x2 x3) = x2.
Let x4 of type ι be given.
Apply H5 with
CD_proj0 x0 x1 (pair_tag x0 x2 x3) = x2.
Assume H6: x1 x4.
Let x5 of type ι → ι → ο be given.
Apply pair_tag_prop_1 with
x0,
x1,
x2,
x3,
CD_proj0 x0 x1 (pair_tag x0 x2 x3),
x4,
λ x6 x7 . x5 x7 x6 leaving 4 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
The subproof is completed by applying H3.
The subproof is completed by applying H7.