Let x0 of type ι be given.
Let x1 of type ο be given.
Assume H1: x0 = 0 ⟶ x1.
Assume H2: x0 = 1 ⟶ x1.
Assume H3:
x0 = Sing 1 ⟶ x1.
Assume H4: x0 = 2 ⟶ x1.
Apply In_Power_ordsucc_cases_impred with
1,
x0,
x1 leaving 3 subgoals.
The subproof is completed by applying H0.
Apply In_Power_1_cases_impred with
x0,
x1 leaving 3 subgoals.
The subproof is completed by applying H5.
Assume H6: x0 = 0.
Apply H1.
The subproof is completed by applying H6.
Assume H6: x0 = 1.
Apply H2.
The subproof is completed by applying H6.
Assume H5: 1 ∈ x0.
Apply In_Power_1_cases_impred with
setminus x0 (Sing 1),
x1 leaving 3 subgoals.
The subproof is completed by applying H6.
Apply H3.
Apply set_ext with
x0,
Sing 1 leaving 2 subgoals.
Let x2 of type ι be given.
Assume H8: x2 ∈ x0.
Apply dneg with
x2 ∈ Sing 1.
Apply EmptyE with
x2.
Apply H7 with
λ x3 x4 . x2 ∈ x3.
Apply setminusI with
x0,
Sing 1,
x2 leaving 2 subgoals.
The subproof is completed by applying H8.
The subproof is completed by applying H9.
Let x2 of type ι be given.
Apply SingE with
1,
x2,
λ x3 x4 . x4 ∈ x0 leaving 2 subgoals.
The subproof is completed by applying H8.
The subproof is completed by applying H5.
Apply H4.
Apply set_ext with
x0,
2 leaving 2 subgoals.
Apply PowerE with
2,
x0.
The subproof is completed by applying H0.
Let x2 of type ι be given.
Assume H8: x2 ∈ 2.
Apply cases_2 with
x2,
λ x3 . x3 ∈ x0 leaving 3 subgoals.
The subproof is completed by applying H8.
Apply setminusE1 with
x0,
Sing 1,
0.
Apply H7 with
λ x3 x4 . 0 ∈ x4.
The subproof is completed by applying In_0_1.
The subproof is completed by applying H5.