Theorem elni

Description: Membership in the class of positive integers. (Contributed by NM, 15-Aug-1995) (New usage is discouraged.)

	Ref	Expression
Assertion	elni	$$⊢A\in 𝑵\leftrightarrow \left(A\in \mathrm{\omega }\wedge A\ne \varnothing \right)$$

Proof

Step	Hyp	Ref	Expression
1		df-ni	$$⊢𝑵=\mathrm{\omega }\setminus \left\{\varnothing \right\}$$
2	1	eleq2i	$$⊢A\in 𝑵\leftrightarrow A\in \left(\mathrm{\omega }\setminus \left\{\varnothing \right\}\right)$$
3		eldifsn	$$⊢A\in \left(\mathrm{\omega }\setminus \left\{\varnothing \right\}\right)\leftrightarrow \left(A\in \mathrm{\omega }\wedge A\ne \varnothing \right)$$
4	2 3	bitri	$$⊢A\in 𝑵\leftrightarrow \left(A\in \mathrm{\omega }\wedge A\ne \varnothing \right)$$