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Theorem elni 9683
 Description: Membership in the class of positive integers. (Contributed by NM, 15-Aug-1995.) (New usage is discouraged.)
Assertion
Ref Expression
elni (𝐴N ↔ (𝐴 ∈ ω ∧ 𝐴 ≠ ∅))

Proof of Theorem elni
StepHypRef Expression
1 df-ni 9679 . . 3 N = (ω ∖ {∅})
21eleq2i 2691 . 2 (𝐴N𝐴 ∈ (ω ∖ {∅}))
3 eldifsn 4308 . 2 (𝐴 ∈ (ω ∖ {∅}) ↔ (𝐴 ∈ ω ∧ 𝐴 ≠ ∅))
42, 3bitri 264 1 (𝐴N ↔ (𝐴 ∈ ω ∧ 𝐴 ≠ ∅))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 196   ∧ wa 384   ∈ wcel 1988   ≠ wne 2791   ∖ cdif 3564  ∅c0 3907  {csn 4168  ωcom 7050  Ncnpi 9651 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-v 3197  df-dif 3570  df-sn 4169  df-ni 9679 This theorem is referenced by:  elni2  9684  0npi  9689  1pi  9690  addclpi  9699  mulclpi  9700  nlt1pi  9713  indpi  9714
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