Let x0 of type ι be given.
Let x1 of type ι be given.
Let x2 of type ι be given.
Assume H0: x0 = x1 ⟶ ∀ x3 : ο . x3.
Assume H1: x0 = x2 ⟶ ∀ x3 : ο . x3.
Assume H2: x1 = x2 ⟶ ∀ x3 : ο . x3.
Let x3 of type ο be given.
Apply H6 with
λ x4 . If_i (x4 = 0) x0 (If_i (x4 = 1) x1 x2).
Apply and3I with
∀ x4 . x4 ∈ 3 ⟶ (λ x5 . If_i (x5 = 0) x0 (If_i (x5 = 1) x1 x2)) x4 ∈ SetAdjoin (UPair x0 x1) x2,
∀ x4 . x4 ∈ 3 ⟶ ∀ x5 . x5 ∈ 3 ⟶ (λ x6 . If_i (x6 = 0) x0 (If_i (x6 = 1) x1 x2)) x4 = (λ x6 . If_i (x6 = 0) x0 (If_i (x6 = 1) x1 x2)) x5 ⟶ x4 = x5,
∀ x4 . x4 ∈ SetAdjoin (UPair x0 x1) x2 ⟶ ∃ x5 . and (x5 ∈ 3) ((λ x6 . If_i (x6 = 0) x0 (If_i (x6 = 1) x1 x2)) x5 = x4) leaving 3 subgoals.
Let x4 of type ι be given.
Assume H7: x4 ∈ 3.
Apply cases_3 with
x4,
λ x5 . (λ x6 . If_i (x6 = 0) x0 (If_i (x6 = 1) x1 x2)) x5 ∈ SetAdjoin (UPair x0 x1) x2 leaving 4 subgoals.
The subproof is completed by applying H7.
Apply binunionI1 with
UPair x0 x1,
Sing x2,
(λ x5 . If_i (x5 = 0) x0 (If_i (x5 = 1) x1 x2)) 0.
Apply L3 with
λ x5 x6 . x6 ∈ UPair x0 x1.
The subproof is completed by applying UPairI1 with x0, x1.
Apply binunionI1 with
UPair x0 x1,
Sing x2,
(λ x5 . If_i (x5 = 0) x0 (If_i (x5 = 1) x1 x2)) 1.
Apply L4 with
λ x5 x6 . x6 ∈ UPair x0 x1.
The subproof is completed by applying UPairI2 with x0, x1.
Apply binunionI2 with
UPair x0 x1,
Sing x2,
(λ x5 . If_i (x5 = 0) x0 (If_i (x5 = 1) x1 x2)) 2.
Apply L5 with
λ x5 x6 . x6 ∈ Sing x2.
The subproof is completed by applying SingI with x2.
Let x4 of type ι be given.
Assume H7: x4 ∈ 3.
Let x5 of type ι be given.
Assume H8: x5 ∈ 3.
Apply cases_3 with
x4,
λ x6 . (λ x7 . If_i (x7 = 0) x0 (If_i (x7 = 1) x1 x2)) x6 = (λ x7 . If_i (x7 = 0) x0 (If_i (x7 = 1) x1 x2)) x5 ⟶ x6 = x5 leaving 4 subgoals.
The subproof is completed by applying H7.
Apply cases_3 with
x5,
λ x6 . ... = ... ⟶ 0 = x6 leaving 4 subgoals.