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Theorem elni 9985
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 9981 . . 3 N = (ω ∖ {∅})
21eleq2i 2869 . 2 (𝐴N𝐴 ∈ (ω ∖ {∅}))
3 eldifsn 4505 . 2 (𝐴 ∈ (ω ∖ {∅}) ↔ (𝐴 ∈ ω ∧ 𝐴 ≠ ∅))
42, 3bitri 267 1 (𝐴N ↔ (𝐴 ∈ ω ∧ 𝐴 ≠ ∅))
Colors of variables: wff setvar class
Syntax hints:  wb 198  wa 385  wcel 2157  wne 2970  cdif 3765  c0 4114  {csn 4367  ωcom 7298  Ncnpi 9953
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-ext 2776
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2785  df-cleq 2791  df-clel 2794  df-nfc 2929  df-ne 2971  df-v 3386  df-dif 3771  df-sn 4368  df-ni 9981
This theorem is referenced by:  elni2  9986  0npi  9991  1pi  9992  addclpi  10001  mulclpi  10002  nlt1pi  10015  indpi  10016
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