Theorem elni 10890

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 10886	. . 3 ⊢ N = (ω ∖ {∅})
2	1	eleq2i 2826	. 2 ⊢ (𝐴 ∈ N ↔ 𝐴 ∈ (ω ∖ {∅}))
3		eldifsn 4762	. 2 ⊢ (𝐴 ∈ (ω ∖ {∅}) ↔ (𝐴 ∈ ω ∧ 𝐴 ≠ ∅))
4	2, 3	bitri 275	1 ⊢ (𝐴 ∈ N ↔ (𝐴 ∈ ω ∧ 𝐴 ≠ ∅))

Syntax hints:   ↔ wb 206   ∧ wa 395   ∈ wcel 2108   ≠ wne 2932   ∖ cdif 3923  ∅c0 4308  {csn 4601  ωcom 7861  Ncnpi 10858

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2714  df-cleq 2727  df-clel 2809  df-ne 2933  df-v 3461  df-dif 3929  df-sn 4602  df-ni 10886

This theorem is referenced by:  elni2  10891  0npi  10896  1pi  10897  addclpi  10906  mulclpi  10907  nlt1pi  10920  indpi  10921