Theorem elni 10632

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 10628	. . 3 ⊢ N = (ω ∖ {∅})
2	1	eleq2i 2830	. 2 ⊢ (𝐴 ∈ N ↔ 𝐴 ∈ (ω ∖ {∅}))
3		eldifsn 4720	. 2 ⊢ (𝐴 ∈ (ω ∖ {∅}) ↔ (𝐴 ∈ ω ∧ 𝐴 ≠ ∅))
4	2, 3	bitri 274	1 ⊢ (𝐴 ∈ N ↔ (𝐴 ∈ ω ∧ 𝐴 ≠ ∅))

Syntax hints:   ↔ wb 205   ∧ wa 396   ∈ wcel 2106   ≠ wne 2943   ∖ cdif 3884  ∅c0 4256  {csn 4561  ωcom 7712  Ncnpi 10600

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1542  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ne 2944  df-v 3434  df-dif 3890  df-sn 4562  df-ni 10628

This theorem is referenced by:  elni2  10633  0npi  10638  1pi  10639  addclpi  10648  mulclpi  10649  nlt1pi  10662  indpi  10663