Let x0 of type ι be given.
Let x1 of type ι be given.
Let x2 of type ι be given.
Apply iffI with
x2 ∈ setsum x0 x1,
or (∃ x3 . and (x3 ∈ x0) (x2 = setsum 0 x3)) (∃ x3 . and (x3 ∈ x1) (x2 = setsum 1 x3)) leaving 2 subgoals.
The subproof is completed by applying pairE with x0, x1, x2.
Apply Inj0_pair_0_eq with
λ x3 x4 : ι → ι . or (∃ x5 . and (x5 ∈ x0) (x2 = x3 x5)) (∃ x5 . and (x5 ∈ x1) (x2 = setsum 1 x5)) ⟶ x2 ∈ setsum x0 x1.
Apply Inj1_pair_1_eq with
λ x3 x4 : ι → ι . or (∃ x5 . and (x5 ∈ x0) (x2 = Inj0 x5)) (∃ x5 . and (x5 ∈ x1) (x2 = x3 x5)) ⟶ x2 ∈ setsum x0 x1.
Assume H0:
or (∃ x3 . and (x3 ∈ x0) (x2 = Inj0 x3)) (∃ x3 . and (x3 ∈ x1) (x2 = Inj1 x3)).
Apply H0 with
x2 ∈ setsum x0 x1 leaving 2 subgoals.
Assume H1:
∃ x3 . and (x3 ∈ x0) (x2 = Inj0 x3).
Apply exandE_i with
λ x3 . x3 ∈ x0,
λ x3 . x2 = Inj0 x3,
x2 ∈ setsum x0 x1 leaving 2 subgoals.
The subproof is completed by applying H1.
Let x3 of type ι be given.
Assume H2: x3 ∈ x0.
Apply H3 with
λ x4 x5 . x5 ∈ setsum x0 x1.
Apply Inj0_setsum with
x0,
x1,
x3.
The subproof is completed by applying H2.
Assume H1:
∃ x3 . and (x3 ∈ x1) (x2 = Inj1 x3).
Apply exandE_i with
λ x3 . x3 ∈ x1,
λ x3 . x2 = Inj1 x3,
x2 ∈ setsum x0 x1 leaving 2 subgoals.
The subproof is completed by applying H1.
Let x3 of type ι be given.
Assume H2: x3 ∈ x1.
Apply H3 with
λ x4 x5 . x5 ∈ setsum x0 x1.
Apply Inj1_setsum with
x0,
x1,
x3.
The subproof is completed by applying H2.