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Proofgold Proof

pf
Let x0 of type ι be given.
Assume H0: x0 ∈ setminus omega 1.
Let x1 of type ο be given.
Assume H1: ∀ x2 . x2 ∈ omega ⟶ x0 = ordsucc x2 ⟶ x1.
Apply setminusE with omega, 1, x0, x1 leaving 2 subgoals.
The subproof is completed by applying H0.
Assume H2: x0 ∈ omega.
Assume H3: nIn x0 1.
Claim L4: nat_p x0
Apply omega_nat_p with x0.
The subproof is completed by applying H2.
Apply nat_inv with x0, x1 leaving 3 subgoals.
The subproof is completed by applying L4.
Assume H5: x0 = 0.
Apply FalseE with x1.
Apply H3.
Apply H5 with λ x2 x3 . x3 ∈ 1.
The subproof is completed by applying In_0_1.
Assume H5: ∃ x2 . and (nat_p x2) (x0 = ordsucc x2).
Apply H5 with x1.
Let x2 of type ι be given.
Assume H6: (λ x3 . and (nat_p x3) (x0 = ordsucc x3)) x2.
Apply H6 with x1.
Assume H7: nat_p x2.
Assume H8: x0 = ordsucc x2.
Apply H1 with x2 leaving 2 subgoals.
Apply nat_p_omega with x2.
The subproof is completed by applying H7.
The subproof is completed by applying H8.
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