Theorem nnssz

Description: Positive integers are a subset of integers. (Contributed by NM, 9-Jan-2002) Reduce dependencies on axioms. (Revised by Steven Nguyen, 29-Nov-2022)

	Ref	Expression
Assertion	nnssz	$$⊢\mathbb{N}\subseteq \mathbb{Z}$$

Proof

Step	Hyp	Ref	Expression
1		nnre	$$⊢x\in \mathbb{N}\to x\in ℝ$$
2		3mix2	$$⊢x\in \mathbb{N}\to \left(x=0\vee x\in \mathbb{N}\vee -x\in \mathbb{N}\right)$$
3		elz	$$⊢x\in \mathbb{Z}\leftrightarrow \left(x\in ℝ\wedge \left(x=0\vee x\in \mathbb{N}\vee -x\in \mathbb{N}\right)\right)$$
4	1 2 3	sylanbrc	$$⊢x\in \mathbb{N}\to x\in \mathbb{Z}$$
5	4	ssriv	$$⊢\mathbb{N}\subseteq \mathbb{Z}$$