Apply nat_complete_ind with
λ x0 . prim3 (ordsucc x0) = x0.
Let x0 of type ι be given.
Apply set_ext with
prim3 (ordsucc x0),
x0 leaving 2 subgoals.
Let x1 of type ι be given.
Apply UnionE_impred with
ordsucc x0,
x1,
x1 ∈ x0 leaving 2 subgoals.
The subproof is completed by applying H2.
Let x2 of type ι be given.
Assume H3: x1 ∈ x2.
Apply nat_ordsucc_trans with
x0,
x2,
x1 leaving 3 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H4.
The subproof is completed by applying H3.
Let x1 of type ι be given.
Assume H2: x1 ∈ x0.
Apply UnionI with
ordsucc x0,
x1,
x0 leaving 2 subgoals.
The subproof is completed by applying H2.
The subproof is completed by applying ordsuccI2 with x0.