Let x0 of type ι be given.

Assume H0: nat_p x0.

Assume H1: 1 ∈ x0.

Let x1 of type ι be given.

Assume H2: ExtendedSNoElt_ x0 x1.

Let x2 of type ι be given.

Assume H3: x2 ∈ x1.

Assume H4: Sing x0 ∈ x2.

Claim L5: Sing x0 ∈ prim3 x1

Apply UnionI with x1, Sing x0, x2 leaving 2 subgoals.

The subproof is completed by applying H4.

The subproof is completed by applying H3.

Apply H2 with Sing x0, False leaving 3 subgoals.

The subproof is completed by applying L5.

Apply not_ordinal_Sing_tagn with x0 leaving 2 subgoals.

The subproof is completed by applying H0.

The subproof is completed by applying H1.

Assume H6: ∃ x3 . and (x3 ∈ x0) (Sing x0 = Sing x3).

Apply H6 with False.

Let x3 of type ι be given.

Assume H7: (λ x4 . and (x4 ∈ x0) (Sing x0 = Sing x4)) x3.

Apply H7 with False.

Assume H8: x3 ∈ x0.

Assume H9: Sing x0 = Sing x3.

Apply In_irref with x3.

Apply Sing_inj with x0, x3, λ x4 x5 . x3 ∈ x4 leaving 2 subgoals.

The subproof is completed by applying H9.

The subproof is completed by applying H8.

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Proofgold Proof