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Theorem elni 10632
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 10628 . . 3 N = (ω ∖ {∅})
21eleq2i 2830 . 2 (𝐴N𝐴 ∈ (ω ∖ {∅}))
3 eldifsn 4720 . 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 396  wcel 2106  wne 2943  cdif 3884  c0 4256  {csn 4561  ωcom 7712  Ncnpi 10600
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709
This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1542  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ne 2944  df-v 3434  df-dif 3890  df-sn 4562  df-ni 10628
This theorem is referenced by:  elni2  10633  0npi  10638  1pi  10639  addclpi  10648  mulclpi  10649  nlt1pi  10662  indpi  10663
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