Theorem elni 10020

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

Assertion

Ref	Expression
elni	⊢ (𝐴 ∈ N ↔ (𝐴 ∈ ω ∧ 𝐴 ≠ ∅))

Proof of Theorem elni

Step	Hyp	Ref	Expression
1		df-ni 10016	. . 3 ⊢ N = (ω ∖ {∅})
2	1	eleq2i 2898	. 2 ⊢ (𝐴 ∈ N ↔ 𝐴 ∈ (ω ∖ {∅}))
3		eldifsn 4538	. 2 ⊢ (𝐴 ∈ (ω ∖ {∅}) ↔ (𝐴 ∈ ω ∧ 𝐴 ≠ ∅))
4	2, 3	bitri 267	1 ⊢ (𝐴 ∈ N ↔ (𝐴 ∈ ω ∧ 𝐴 ≠ ∅))

Syntax hints:   ↔ wb 198   ∧ wa 386   ∈ wcel 2164   ≠ wne 2999   ∖ cdif 3795  ∅c0 4146  {csn 4399  ωcom 7331  Ncnpi 9988

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-ext 2803

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-v 3416  df-dif 3801  df-sn 4400  df-ni 10016

This theorem is referenced by:  elni2  10021  0npi  10026  1pi  10027  addclpi  10036  mulclpi  10037  nlt1pi  10050  indpi  10051