Let x0 of type ι be given.
Let x1 of type ι be given.
Apply H0 with
λ x2 x3 . x0 ∈ x2.
The subproof is completed by applying ordsuccI2 with x0.
Apply ordsuccE with
x1,
x0,
x0 = x1 leaving 3 subgoals.
The subproof is completed by applying L1.
Assume H2: x0 ∈ x1.
Apply H0 with
λ x2 x3 . x1 ∈ x3.
The subproof is completed by applying ordsuccI2 with x1.
Apply ordsuccE with
x0,
x1,
x0 = x1 leaving 3 subgoals.
The subproof is completed by applying L3.
Assume H4: x1 ∈ x0.
Apply FalseE with
x0 = x1.
Apply In_no2cycle with
x0,
x1 leaving 2 subgoals.
The subproof is completed by applying H2.
The subproof is completed by applying H4.
Assume H4: x1 = x0.
Let x2 of type ι → ι → ο be given.
The subproof is completed by applying H4 with λ x3 x4 . x2 x4 x3.
Assume H2: x0 = x1.
The subproof is completed by applying H2.