Theorem nnz

Description: A positive integer is an integer. (Contributed by NM, 9-May-2004) Reduce dependencies on axioms. (Revised by Steven Nguyen, 29-Nov-2022)

	Ref	Expression
Assertion	nnz	$$⊢N\in \mathbb{N}\to N\in \mathbb{Z}$$

Proof

Step	Hyp	Ref	Expression
1		nnre	$$⊢N\in \mathbb{N}\to N\in ℝ$$
2		3mix2	$$⊢N\in \mathbb{N}\to \left(N=0\vee N\in \mathbb{N}\vee -N\in \mathbb{N}\right)$$
3		elz	$$⊢N\in \mathbb{Z}\leftrightarrow \left(N\in ℝ\wedge \left(N=0\vee N\in \mathbb{N}\vee -N\in \mathbb{N}\right)\right)$$
4	1 2 3	sylanbrc	$$⊢N\in \mathbb{N}\to N\in \mathbb{Z}$$