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Proofgold Proof

pf
Apply nat_ind with λ x0 . ∀ x1 . x1 ∈ x0 ⟶ nat_p x1 leaving 2 subgoals.
Let x0 of type ι be given.
Assume H0: x0 ∈ 0.
Apply FalseE with nat_p x0.
Apply EmptyE with x0.
The subproof is completed by applying H0.
Let x0 of type ι be given.
Assume H0: nat_p x0.
Assume H1: ∀ x1 . x1 ∈ x0 ⟶ nat_p x1.
Let x1 of type ι be given.
Assume H2: x1 ∈ ordsucc x0.
Apply ordsuccE with x0, x1, nat_p x1 leaving 3 subgoals.
The subproof is completed by applying H2.
Assume H3: x1 ∈ x0.
Apply H1 with x1.
The subproof is completed by applying H3.
Assume H3: x1 = x0.
Apply H3 with λ x2 x3 . nat_p x3.
The subproof is completed by applying H0.
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