Let x0 of type ι be given.
Apply H0 with
∃ x1 . and (x1 ∈ x0) (x0 = Sing x1).
Let x1 of type ι → ι be given.
Apply bijE with
u1,
x0,
x1,
∃ x2 . and (x2 ∈ x0) (x0 = Sing x2) leaving 2 subgoals.
The subproof is completed by applying H1.
Assume H2:
∀ x2 . x2 ∈ u1 ⟶ x1 x2 ∈ x0.
Assume H3:
∀ x2 . x2 ∈ u1 ⟶ ∀ x3 . x3 ∈ u1 ⟶ x1 x2 = x1 x3 ⟶ x2 = x3.
Assume H4:
∀ x2 . x2 ∈ x0 ⟶ ∃ x3 . and (x3 ∈ u1) (x1 x3 = x2).
Let x2 of type ο be given.
Assume H5:
∀ x3 . and (x3 ∈ x0) (x0 = Sing x3) ⟶ x2.
Apply H5 with
x1 0.
Apply andI with
x1 0 ∈ x0,
x0 = Sing (x1 0) leaving 2 subgoals.
Apply H2 with
0.
The subproof is completed by applying In_0_1.
Apply set_ext with
x0,
Sing (x1 0) leaving 2 subgoals.
Let x3 of type ι be given.
Assume H6: x3 ∈ x0.
Apply H4 with
x3,
x3 ∈ Sing (x1 0) leaving 2 subgoals.
The subproof is completed by applying H6.
Let x4 of type ι be given.
Assume H7:
(λ x5 . and (x5 ∈ u1) (x1 x5 = x3)) x4.
Apply H7 with
x3 ∈ Sing (x1 0).
Assume H8: x4 ∈ 1.
Apply cases_1 with
x4,
λ x5 . x1 x5 = x3 ⟶ x3 ∈ Sing (x1 0) leaving 2 subgoals.
The subproof is completed by applying H8.
Assume H9: x1 0 = x3.
Apply H9 with
λ x5 x6 . x3 ∈ Sing x6.
The subproof is completed by applying SingI with x3.
Let x3 of type ι be given.
Assume H6:
x3 ∈ Sing (x1 0).
Apply SingE with
x1 0,
x3,
λ x4 x5 . x5 ∈ x0 leaving 2 subgoals.
The subproof is completed by applying H6.
Apply H2 with
0.
The subproof is completed by applying In_0_1.