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Theorem elni 6921
Description: Membership in the class of positive integers. (Contributed by NM, 15-Aug-1995.)
Assertion
Ref Expression
elni  |-  ( A  e.  N.  <->  ( A  e.  om  /\  A  =/=  (/) ) )

Proof of Theorem elni
StepHypRef Expression
1 df-ni 6917 . . 3  |-  N.  =  ( om  \  { (/) } )
21eleq2i 2155 . 2  |-  ( A  e.  N.  <->  A  e.  ( om  \  { (/) } ) )
3 eldifsn 3573 . 2  |-  ( A  e.  ( om  \  { (/)
} )  <->  ( A  e.  om  /\  A  =/=  (/) ) )
42, 3bitri 183 1  |-  ( A  e.  N.  <->  ( A  e.  om  /\  A  =/=  (/) ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 103    <-> wb 104    e. wcel 1439    =/= wne 2256    \ cdif 2997   (/)c0 3287   {csn 3450   omcom 4418   N.cnpi 6885
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071
This theorem depends on definitions:  df-bi 116  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ne 2257  df-v 2622  df-dif 3002  df-sn 3456  df-ni 6917
This theorem is referenced by:  0npi  6926  elni2  6927  1pi  6928  addclpi  6940  mulclpi  6941  nlt1pig  6954  indpi  6955  nqnq0pi  7051  prarloclemcalc  7115
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