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Proofgold Proof

pf
Let x0 of type ιο be given.
Assume H0: ∀ x1 . x1omegax0 x1.
Assume H1: ∀ x1 . x1omegax0 (minus_SNo x1).
Let x1 of type ι be given.
Assume H2: x1int.
Apply binunionE with omega, {minus_SNo x2|x2 ∈ omega}, x1, x0 x1 leaving 3 subgoals.
The subproof is completed by applying H2.
The subproof is completed by applying H0 with x1.
Assume H3: x1prim5 omega minus_SNo.
Apply ReplE_impred with omega, minus_SNo, x1, x0 x1 leaving 2 subgoals.
The subproof is completed by applying H3.
Let x2 of type ι be given.
Assume H4: x2omega.
Assume H5: x1 = minus_SNo x2.
Apply H5 with λ x3 x4 . x0 x4.
Apply H1 with x2.
The subproof is completed by applying H4.