Let x0 of type ι be given.
Let x1 of type ι be given.
Let x2 of type ο be given.
Assume H1:
x1 ∈ prim4 x0 ⟶ x2.
Apply PowerE with
ordsucc x0,
x1.
The subproof is completed by applying H0.
Apply xm with
x0 ∈ x1,
x2 leaving 2 subgoals.
Assume H4: x0 ∈ x1.
Apply H2 leaving 2 subgoals.
The subproof is completed by applying H4.
Apply PowerI with
x0,
setminus x1 (Sing x0).
Let x3 of type ι be given.
Apply setminusE with
x1,
Sing x0,
x3,
x3 ∈ x0 leaving 2 subgoals.
The subproof is completed by applying H5.
Assume H6: x3 ∈ x1.
Apply ordsuccE with
x0,
x3,
x3 ∈ x0 leaving 3 subgoals.
Apply L3 with
x3.
The subproof is completed by applying H6.
Assume H8: x3 ∈ x0.
The subproof is completed by applying H8.
Assume H8: x3 = x0.
Apply FalseE with
x3 ∈ x0.
Apply H7.
Apply H8 with
λ x4 x5 . x5 ∈ Sing x0.
The subproof is completed by applying SingI with x0.
Apply H1.
Apply PowerI with
x0,
x1.
Let x3 of type ι be given.
Assume H5: x3 ∈ x1.
Apply ordsuccE with
x0,
x3,
x3 ∈ x0 leaving 3 subgoals.
Apply L3 with
x3.
The subproof is completed by applying H5.
Assume H6: x3 ∈ x0.
The subproof is completed by applying H6.
Assume H6: x3 = x0.
Apply FalseE with
x3 ∈ x0.
Apply H4.
Apply H6 with
λ x4 x5 . x4 ∈ x1.
The subproof is completed by applying H5.