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Theorem elni 9303
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 9299 . . 3  |-  N.  =  ( om  \  { (/) } )
21eleq2i 2501 . 2  |-  ( A  e.  N.  <->  A  e.  ( om  \  { (/) } ) )
3 eldifsn 4123 . 2  |-  ( A  e.  ( om  \  { (/)
} )  <->  ( A  e.  om  /\  A  =/=  (/) ) )
42, 3bitri 253 1  |-  ( A  e.  N.  <->  ( A  e.  om  /\  A  =/=  (/) ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 188    /\ wa 371    e. wcel 1869    =/= wne 2619    \ cdif 3434   (/)c0 3762   {csn 3997   omcom 6704   N.cnpi 9271
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401
This theorem depends on definitions:  df-bi 189  df-an 373  df-tru 1441  df-ex 1661  df-nf 1665  df-sb 1788  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ne 2621  df-v 3084  df-dif 3440  df-sn 3998  df-ni 9299
This theorem is referenced by:  elni2  9304  0npi  9309  1pi  9310  addclpi  9319  mulclpi  9320  nlt1pi  9333  indpi  9334
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