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Theorem elni 8713
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 8709 . . 3  |-  N.  =  ( om  \  { (/) } )
21eleq2i 2472 . 2  |-  ( A  e.  N.  <->  A  e.  ( om  \  { (/) } ) )
3 eldifsn 3891 . 2  |-  ( A  e.  ( om  \  { (/)
} )  <->  ( A  e.  om  /\  A  =/=  (/) ) )
42, 3bitri 241 1  |-  ( A  e.  N.  <->  ( A  e.  om  /\  A  =/=  (/) ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359    e. wcel 1721    =/= wne 2571    \ cdif 3281   (/)c0 3592   {csn 3778   omcom 4808   N.cnpi 8679
This theorem is referenced by:  elni2  8714  0npi  8719  1pi  8720  addclpi  8729  mulclpi  8730  nlt1pi  8743  indpi  8744
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-v 2922  df-dif 3287  df-sn 3784  df-ni 8709
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