Theorem xnn0nnd

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

	Ref	Expression
Hypotheses	xnn0nnd.1	⊢ ( 𝜑 → 𝑁 ∈ ℕ0* )
	xnn0nnd.2	⊢ ( 𝜑 → 𝑁 ∈ ℝ )
	xnn0nnd.3	⊢ ( 𝜑 → 0 < 𝑁 )
Assertion	xnn0nnd	⊢ ( 𝜑 → 𝑁 ∈ ℕ )

Proof

Step	Hyp	Ref	Expression
1		xnn0nnd.1	⊢ ( 𝜑 → 𝑁 ∈ ℕ0* )
2		xnn0nnd.2	⊢ ( 𝜑 → 𝑁 ∈ ℝ )
3		xnn0nnd.3	⊢ ( 𝜑 → 0 < 𝑁 )
4	1 2	xnn0nn0d	⊢ ( 𝜑 → 𝑁 ∈ ℕ0 )
5		elnnnn0b	⊢ ( 𝑁 ∈ ℕ ↔ ( 𝑁 ∈ ℕ0 ∧ 0 < 𝑁 ) )
6	4 3 5	sylanbrc	⊢ ( 𝜑 → 𝑁 ∈ ℕ )