Let x0 of type ι be given.
Apply H0 with
ordinal (ordsucc x0).
Assume H2:
∀ x1 . x1 ∈ x0 ⟶ TransSet x1.
Apply andI with
TransSet (ordsucc x0),
∀ x1 . x1 ∈ ordsucc x0 ⟶ TransSet x1 leaving 2 subgoals.
Apply TransSet_ordsucc with
x0.
The subproof is completed by applying H1.
Let x1 of type ι be given.
Apply ordsuccE with
x0,
x1,
TransSet x1 leaving 3 subgoals.
The subproof is completed by applying H3.
Assume H4: x1 ∈ x0.
Apply H2 with
x1.
The subproof is completed by applying H4.
Assume H4: x1 = x0.
Apply H4 with
λ x2 x3 . TransSet x3.
The subproof is completed by applying H1.