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 ∈ N ↔ ( 𝐴 ∈ ω ∧ 𝐴 ≠ ∅ ) )

Proof

Step	Hyp	Ref	Expression
1		df-ni	⊢ N = ( ω ∖ { ∅ } )
2	1	eleq2i	⊢ ( 𝐴 ∈ N ↔ 𝐴 ∈ ( ω ∖ { ∅ } ) )
3		eldifsn	⊢ ( 𝐴 ∈ ( ω ∖ { ∅ } ) ↔ ( 𝐴 ∈ ω ∧ 𝐴 ≠ ∅ ) )
4	2 3	bitri	⊢ ( 𝐴 ∈ N ↔ ( 𝐴 ∈ ω ∧ 𝐴 ≠ ∅ ) )