Let x0 of type ι → ο be given.
Assume H0:
∀ x1 . nat_p x1 ⟶ (∀ x2 . x2 ∈ x1 ⟶ x0 x2) ⟶ x0 x1.
Claim L1:
∀ x1 . nat_p x1 ⟶ ∀ x2 . x2 ∈ x1 ⟶ x0 x2Apply nat_ind with
λ x1 . ∀ x2 . x2 ∈ x1 ⟶ x0 x2 leaving 2 subgoals.
Let x1 of type ι be given.
Assume H1: x1 ∈ 0.
Apply FalseE with
x0 x1.
Apply EmptyE with
x1.
The subproof is completed by applying H1.
Let x1 of type ι be given.
Assume H2: ∀ x2 . x2 ∈ x1 ⟶ x0 x2.
Let x2 of type ι be given.
Apply ordsuccE with
x1,
x2,
x0 x2 leaving 3 subgoals.
The subproof is completed by applying H3.
Assume H4: x2 ∈ x1.
Apply H2 with
x2.
The subproof is completed by applying H4.
Assume H4: x2 = x1.
Apply H4 with
λ x3 x4 . x0 x4.
Apply H0 with
x1 leaving 2 subgoals.
The subproof is completed by applying H1.
The subproof is completed by applying H2.
Let x1 of type ι be given.
Apply H0 with
x1 leaving 2 subgoals.
The subproof is completed by applying H2.
Apply L1 with
x1.
The subproof is completed by applying H2.