Let x0 of type ι be given.
Let x1 of type ι → ι be given.
Assume H1:
∀ x2 . x2 ∈ x0 ⟶ x1 x2 ∈ prim4 1.
Apply PowerI with
1,
lam x0 (λ x2 . x1 x2).
Let x2 of type ι be given.
Assume H2:
x2 ∈ lam x0 (λ x3 . x1 x3).
Claim L3: x2 = 0
Apply and3E with
setsum (proj0 x2) (proj1 x2) = x2,
proj0 x2 ∈ x0,
proj1 x2 ∈ x1 (proj0 x2),
x2 = 0 leaving 2 subgoals.
Apply Sigma_eta_proj0_proj1 with
x0,
x1,
x2.
The subproof is completed by applying H2.
Assume H4:
proj0 x2 ∈ x0.
Apply SingE with
0,
proj0 x2.
Apply Subq_1_Sing0 with
proj0 x2.
Apply PowerE with
1,
x0,
proj0 x2 leaving 2 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H4.
Apply SingE with
0,
proj1 x2.
Apply Subq_1_Sing0 with
proj1 x2.
Apply PowerE with
1,
x1 (proj0 x2),
proj1 x2 leaving 2 subgoals.
Apply H1 with
proj0 x2.
The subproof is completed by applying H4.
The subproof is completed by applying H5.
Apply H3 with
λ x3 x4 . x3 = 0.
Apply L6 with
λ x3 x4 . setsum x4 (proj1 x2) = 0.
Apply L7 with
λ x3 x4 . setsum 0 x4 = 0.
The subproof is completed by applying setsum_0_0.
Apply L3 with
λ x3 x4 . x4 ∈ 1.
The subproof is completed by applying In_0_1.