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Theorem elni 9044
Description: Membership in the class of positive integers. (Contributed by NM, 15-Aug-1995.) (New usage is discouraged.)
Assertion
Ref Expression
elni  |-  ( A  e.  N.  <->  ( A  e.  om  /\  A  =/=  (/) ) )

Proof of Theorem elni
StepHypRef Expression
1 df-ni 9040 . . 3  |-  N.  =  ( om  \  { (/) } )
21eleq2i 2506 . 2  |-  ( A  e.  N.  <->  A  e.  ( om  \  { (/) } ) )
3 eldifsn 3999 . 2  |-  ( A  e.  ( om  \  { (/)
} )  <->  ( A  e.  om  /\  A  =/=  (/) ) )
42, 3bitri 249 1  |-  ( A  e.  N.  <->  ( A  e.  om  /\  A  =/=  (/) ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    e. wcel 1756    =/= wne 2605    \ cdif 3324   (/)c0 3636   {csn 3876   omcom 6475   N.cnpi 9010
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-v 2973  df-dif 3330  df-sn 3877  df-ni 9040
This theorem is referenced by:  elni2  9045  0npi  9050  1pi  9051  addclpi  9060  mulclpi  9061  nlt1pi  9074  indpi  9075
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