Theorem elni 10857

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 10853	. . 3 ⊢ N = (ω ∖ {∅})
2	1	eleq2i 2861	. 2 ⊢ (𝐴 ∈ N ↔ 𝐴 ∈ (ω ∖ {∅}))
3		eldifsn 4755	. 2 ⊢ (𝐴 ∈ (ω ∖ {∅}) ↔ (𝐴 ∈ ω ∧ 𝐴 ≠ ∅))
4	2, 3	bitri 278	1 ⊢ (𝐴 ∈ N ↔ (𝐴 ∈ ω ∧ 𝐴 ≠ ∅))

Syntax hints:   ↔ wb 209   ∧ wa 400   ∈ wcel 2149   ≠ wne 2964   ∖ cdif 3910  ∅c0 4294  {csn 4591  ωcom 7858  Ncnpi 10825

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1570  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-ne 2965  df-v 3465  df-dif 3916  df-sn 4592  df-ni 10853

This theorem is referenced by:  elni2  10858  0npi  10863  1pi  10864  addclpi  10873  mulclpi  10874  nlt1pi  10887  indpi  10888