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Proofgold Proof

pf
Let x0 of type ι be given.
Assume H0: nat_p x0.
Assume H1: 1x0.
Assume H2: TransSet (Sing x0).
Claim L3: 0Sing x0
Apply H2 with x0, 0 leaving 2 subgoals.
The subproof is completed by applying SingI with x0.
Apply nat_trans with x0, 1, 0 leaving 3 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
The subproof is completed by applying In_0_1.
Apply In_no2cycle with 0, 1 leaving 2 subgoals.
The subproof is completed by applying In_0_1.
Apply SingE with x0, 0, λ x1 x2 . 1x2 leaving 2 subgoals.
The subproof is completed by applying L3.
The subproof is completed by applying H1.