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Theorem elni 9900
 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 9896 . . 3 N = (ω ∖ {∅})
21eleq2i 2842 . 2 (𝐴N𝐴 ∈ (ω ∖ {∅}))
3 eldifsn 4453 . 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 382   ∈ wcel 2145   ≠ wne 2943   ∖ cdif 3720  ∅c0 4063  {csn 4316  ωcom 7212  Ncnpi 9868 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-v 3353  df-dif 3726  df-sn 4317  df-ni 9896 This theorem is referenced by:  elni2  9901  0npi  9906  1pi  9907  addclpi  9916  mulclpi  9917  nlt1pi  9930  indpi  9931
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