Let x0 of type ι be given.
Let x1 of type ι be given.
Apply nat_inv with
x1,
or (x1 = 0) (∃ x2 . and (x2 ∈ x0) (x1 = ordsucc x2)) leaving 3 subgoals.
Apply nat_p_trans with
ordsucc x0,
x1 leaving 2 subgoals.
Apply nat_ordsucc with
x0.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
The subproof is completed by applying orIL with
x1 = 0,
∃ x2 . and (x2 ∈ x0) (x1 = ordsucc x2).
Apply H2 with
or (x1 = 0) (∃ x2 . and (x2 ∈ x0) (x1 = ordsucc x2)).
Let x2 of type ι be given.
Apply H3 with
or (x1 = 0) (∃ x3 . and (x3 ∈ x0) (x1 = ordsucc x3)).
Apply orIR with
x1 = 0,
∃ x3 . and (x3 ∈ x0) (x1 = ordsucc x3).
Let x3 of type ο be given.
Assume H6:
∀ x4 . and (x4 ∈ x0) (x1 = ordsucc x4) ⟶ x3.
Apply H6 with
x2.
Apply andI with
x2 ∈ x0,
x1 = ordsucc x2 leaving 2 subgoals.
Apply ordsuccE with
x0,
x1,
x2 ∈ x0 leaving 3 subgoals.
The subproof is completed by applying H1.
Assume H7: x1 ∈ x0.
Apply nat_trans with
x0,
x1,
x2 leaving 3 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H7.
Apply H5 with
λ x4 x5 . x2 ∈ x5.
The subproof is completed by applying ordsuccI2 with x2.
Assume H7: x1 = x0.
Apply H7 with
λ x4 x5 . x2 ∈ x4.
Apply H5 with
λ x4 x5 . x2 ∈ x5.
The subproof is completed by applying ordsuccI2 with x2.
The subproof is completed by applying H5.