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Theorem elni 10901
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 10897 . . 3 N = (ω ∖ {∅})
21eleq2i 2817 . 2 (𝐴N𝐴 ∈ (ω ∖ {∅}))
3 eldifsn 4792 . 2 (𝐴 ∈ (ω ∖ {∅}) ↔ (𝐴 ∈ ω ∧ 𝐴 ≠ ∅))
42, 3bitri 274 1 (𝐴N ↔ (𝐴 ∈ ω ∧ 𝐴 ≠ ∅))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 394  wcel 2098  wne 2929  cdif 3941  c0 4322  {csn 4630  ωcom 7871  Ncnpi 10869
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696
This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-ne 2930  df-v 3463  df-dif 3947  df-sn 4631  df-ni 10897
This theorem is referenced by:  elni2  10902  0npi  10907  1pi  10908  addclpi  10917  mulclpi  10918  nlt1pi  10931  indpi  10932
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