Let x0 of type ι be given.
Let x1 of type ι be given.
Let x2 of type ι be given.
Apply pairE with
x0,
x1,
setsum 1 x2,
x2 ∈ x1 leaving 3 subgoals.
The subproof is completed by applying H0.
Apply Inj0_pair_0_eq with
λ x3 x4 : ι → ι . (∃ x5 . and (x5 ∈ x0) (setsum 1 x2 = x3 x5)) ⟶ x2 ∈ x1.
Apply Inj1_pair_1_eq with
λ x3 x4 : ι → ι . (∃ x5 . and (x5 ∈ x0) (x3 x2 = Inj0 x5)) ⟶ x2 ∈ x1.
Apply FalseE with
x2 ∈ x1.
Apply exandE_i with
λ x3 . x3 ∈ x0,
λ x3 . Inj1 x2 = Inj0 x3,
False leaving 2 subgoals.
The subproof is completed by applying H1.
Let x3 of type ι be given.
Assume H2: x3 ∈ x0.
Apply Inj0_Inj1_neq with
x3,
x2.
Let x4 of type ι → ι → ο be given.
The subproof is completed by applying H3 with λ x5 x6 . x4 x6 x5.
Apply Inj1_pair_1_eq with
λ x3 x4 : ι → ι . (∃ x5 . and (x5 ∈ x1) (x3 x2 = x3 x5)) ⟶ x2 ∈ x1.
Apply exandE_i with
λ x3 . x3 ∈ x1,
λ x3 . Inj1 x2 = Inj1 x3,
x2 ∈ x1 leaving 2 subgoals.
The subproof is completed by applying H1.
Let x3 of type ι be given.
Assume H2: x3 ∈ x1.
Apply Inj1_inj with
x2,
x3,
λ x4 x5 . x5 ∈ x1 leaving 2 subgoals.
The subproof is completed by applying H3.
The subproof is completed by applying H2.