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Theorem elni 7324
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 7320 . . 3  |-  N.  =  ( om  \  { (/) } )
21eleq2i 2255 . 2  |-  ( A  e.  N.  <->  A  e.  ( om  \  { (/) } ) )
3 eldifsn 3733 . 2  |-  ( A  e.  ( om  \  { (/)
} )  <->  ( A  e.  om  /\  A  =/=  (/) ) )
42, 3bitri 184 1  |-  ( A  e.  N.  <->  ( A  e.  om  /\  A  =/=  (/) ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 104    <-> wb 105    e. wcel 2159    =/= wne 2359    \ cdif 3140   (/)c0 3436   {csn 3606   omcom 4603   N.cnpi 7288
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2170
This theorem depends on definitions:  df-bi 117  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2175  df-cleq 2181  df-clel 2184  df-nfc 2320  df-ne 2360  df-v 2753  df-dif 3145  df-sn 3612  df-ni 7320
This theorem is referenced by:  0npi  7329  elni2  7330  1pi  7331  addclpi  7343  mulclpi  7344  nlt1pig  7357  indpi  7358  nqnq0pi  7454  prarloclemcalc  7518
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