Let x0 of type ι be given.
Apply H1 with
False.
Let x1 of type ι → ι be given.
Apply H2 with
False.
Assume H3:
∀ x2 . x2 ∈ ordsucc x0 ⟶ x1 x2 ∈ x0.
Assume H4:
∀ x2 . x2 ∈ ordsucc x0 ⟶ ∀ x3 . x3 ∈ ordsucc x0 ⟶ x1 x2 = x1 x3 ⟶ x2 = x3.
Apply PigeonHole_nat with
x0,
x1 leaving 3 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H3.
The subproof is completed by applying H4.