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Theorem elni 10379
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 10375 . . 3 N = (ω ∖ {∅})
21eleq2i 2825 . 2 (𝐴N𝐴 ∈ (ω ∖ {∅}))
3 eldifsn 4676 . 2 (𝐴 ∈ (ω ∖ {∅}) ↔ (𝐴 ∈ ω ∧ 𝐴 ≠ ∅))
42, 3bitri 278 1 (𝐴N ↔ (𝐴 ∈ ω ∧ 𝐴 ≠ ∅))
Colors of variables: wff setvar class
Syntax hints:  wb 209  wa 399  wcel 2114  wne 2935  cdif 3841  c0 4212  {csn 4517  ωcom 7602  Ncnpi 10347
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-ext 2711
This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1545  df-ex 1787  df-sb 2075  df-clab 2718  df-cleq 2731  df-clel 2812  df-ne 2936  df-v 3401  df-dif 3847  df-sn 4518  df-ni 10375
This theorem is referenced by:  elni2  10380  0npi  10385  1pi  10386  addclpi  10395  mulclpi  10396  nlt1pi  10409  indpi  10410
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