Let x0 of type ι be given.
Let x1 of type ι be given.
Let x2 of type ι be given.
Assume H0:
∀ x3 . x3 ∈ x0 ⟶ or (x3 = x1) (x3 = x2).
Let x3 of type ο be given.
Assume H1:
∀ x4 : ι → ι . inj x0 u2 x4 ⟶ x3.
Apply H1 with
λ x4 . If_i (x4 = x1) 0 u1.
Apply andI with
∀ x4 . x4 ∈ x0 ⟶ (λ x5 . If_i (x5 = x1) 0 u1) x4 ∈ u2,
∀ x4 . x4 ∈ x0 ⟶ ∀ x5 . x5 ∈ x0 ⟶ (λ x6 . If_i (x6 = x1) 0 u1) x4 = (λ x6 . If_i (x6 = x1) 0 u1) x5 ⟶ x4 = x5 leaving 2 subgoals.
Let x4 of type ι be given.
Assume H2: x4 ∈ x0.
Apply If_i_or with
x4 = x1,
0,
u1,
If_i (x4 = x1) 0 u1 ∈ u2 leaving 2 subgoals.
Assume H3:
If_i (x4 = x1) 0 u1 = 0.
Apply H3 with
λ x5 x6 . x6 ∈ u2.
The subproof is completed by applying In_0_2.
Apply H3 with
λ x5 x6 . x6 ∈ u2.
The subproof is completed by applying In_1_2.
Let x4 of type ι be given.
Assume H2: x4 ∈ x0.
Let x5 of type ι be given.
Assume H3: x5 ∈ x0.
Apply xm with
x4 = x1,
(λ x6 . If_i (x6 = x1) 0 u1) x4 = (λ x6 . If_i (x6 = x1) 0 u1) x5 ⟶ x4 = x5 leaving 2 subgoals.
Assume H4: x4 = x1.
Apply xm with
x5 = x1,
(λ x6 . If_i (x6 = x1) 0 u1) x4 = (λ x6 . If_i (x6 = x1) 0 u1) x5 ⟶ x4 = x5 leaving 2 subgoals.
Assume H5: x5 = x1.
Assume H6:
(λ x6 . If_i (x6 = x1) 0 u1) x4 = (λ x6 . If_i (x6 = x1) 0 u1) x5.
Apply H5 with
λ x6 x7 . x4 = x7.
The subproof is completed by applying H4.
Assume H5: x5 = x1 ⟶ ∀ x6 : ο . x6.
Apply If_i_1 with
x4 = x1,
0,
u1,
λ x6 x7 . x7 = If_i (x5 = x1) 0 u1 ⟶ x4 = x5 leaving 2 subgoals.
The subproof is completed by applying H4.
Apply If_i_0 with
x5 = x1,
0,
u1,
λ x6 x7 . 0 = x7 ⟶ x4 = x5 leaving 2 subgoals.
The subproof is completed by applying H5.
Apply FalseE with
x4 = x5.
Apply neq_0_1.
The subproof is completed by applying H6.
Assume H4: x4 = x1 ⟶ ∀ x6 : ο . x6.
Apply xm with
x5 = x1,
(λ x6 . If_i (x6 = x1) 0 u1) x4 = (λ x6 . If_i (x6 = x1) 0 u1) x5 ⟶ x4 = x5 leaving 2 subgoals.
Assume H5: x5 = x1.
Apply If_i_0 with
x4 = x1,
0,
u1,
λ x6 x7 . x7 = If_i (x5 = x1) 0 ... ⟶ x4 = x5 leaving 2 subgoals.