Theorem xnn0nn0d

Description: Conditions for an extended nonnegative integer to be a nonnegative integer. (Contributed by Thierry Arnoux, 26-Oct-2025)

	Ref	Expression
Hypotheses	xnn0nnd.1	⊢ ( 𝜑 → 𝑁 ∈ ℕ0* )
	xnn0nnd.2	⊢ ( 𝜑 → 𝑁 ∈ ℝ )
Assertion	xnn0nn0d	⊢ ( 𝜑 → 𝑁 ∈ ℕ0 )

Proof

Step	Hyp	Ref	Expression
1		xnn0nnd.1	⊢ ( 𝜑 → 𝑁 ∈ ℕ0* )
2		xnn0nnd.2	⊢ ( 𝜑 → 𝑁 ∈ ℝ )
3		elxnn0	⊢ ( 𝑁 ∈ ℕ0* ↔ ( 𝑁 ∈ ℕ0 ∨ 𝑁 = +∞ ) )
4	1 3	sylib	⊢ ( 𝜑 → ( 𝑁 ∈ ℕ0 ∨ 𝑁 = +∞ ) )
5	2	renepnfd	⊢ ( 𝜑 → 𝑁 ≠ +∞ )
6	5	neneqd	⊢ ( 𝜑 → ¬ 𝑁 = +∞ )
7	4 6	olcnd	⊢ ( 𝜑 → 𝑁 ∈ ℕ0 )