Let x0 of type ι be given.
Let x1 of type ι → ι be given.
Let x2 of type ι → ι be given.
Assume H0: ∀ x3 . x3 ∈ x0 ⟶ x2 x3 ∈ x1 x3.
Apply PiI with
x0,
λ x3 . x1 x3,
lam x0 (λ x3 . x2 x3) leaving 2 subgoals.
Let x3 of type ι be given.
Assume H1:
x3 ∈ lam x0 (λ x4 . x2 x4).
Claim L2:
∃ x4 . and (x4 ∈ x0) (∃ x5 . and (x5 ∈ x2 x4) (x3 = setsum x4 x5))Apply lamE with
x0,
x2,
x3.
The subproof is completed by applying H1.
Apply exandE_i with
λ x4 . x4 ∈ x0,
λ x4 . ∃ x5 . and (x5 ∈ x2 x4) (x3 = setsum x4 x5),
and (pair_p x3) (ap x3 0 ∈ x0) leaving 2 subgoals.
The subproof is completed by applying L2.
Let x4 of type ι be given.
Assume H3: x4 ∈ x0.
Assume H4:
∃ x5 . and (x5 ∈ x2 x4) (x3 = setsum x4 x5).
Apply exandE_i with
λ x5 . x5 ∈ x2 x4,
λ x5 . x3 = setsum x4 x5,
and (pair_p x3) (ap x3 0 ∈ x0) leaving 2 subgoals.
The subproof is completed by applying H4.
Let x5 of type ι be given.
Assume H5: x5 ∈ x2 x4.
Apply andI with
pair_p x3,
ap x3 0 ∈ x0 leaving 2 subgoals.
Apply H6 with
λ x6 x7 . pair_p x7.
The subproof is completed by applying pair_p_I with x4, x5.
Apply H6 with
λ x6 x7 . ap x7 0 ∈ x0.
Apply pair_ap_0 with
x4,
x5,
λ x6 x7 . x7 ∈ x0.
The subproof is completed by applying H3.
Let x3 of type ι be given.
Assume H1: x3 ∈ x0.
Apply beta with
x0,
x2,
x3,
λ x4 x5 . x5 ∈ x1 x3 leaving 2 subgoals.
The subproof is completed by applying H1.
Apply H0 with
x3.
The subproof is completed by applying H1.