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Theorem pinn 8502
Description: A positive integer is a natural number. (Contributed by NM, 15-Aug-1995.) (New usage is discouraged.)
Assertion
Ref Expression
pinn  |-  ( A  e.  N.  ->  A  e.  om )

Proof of Theorem pinn
StepHypRef Expression
1 df-ni 8496 . . 3  |-  N.  =  ( om  \  { (/) } )
2 difss 3303 . . 3  |-  ( om 
\  { (/) } ) 
C_  om
31, 2eqsstri 3208 . 2  |-  N.  C_  om
43sseli 3176 1  |-  ( A  e.  N.  ->  A  e.  om )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1684    \ cdif 3149   (/)c0 3455   {csn 3640   omcom 4656   N.cnpi 8466
This theorem is referenced by:  pion  8503  piord  8504  mulidpi  8510  addclpi  8516  mulclpi  8517  addcompi  8518  addasspi  8519  mulcompi  8520  mulasspi  8521  distrpi  8522  addcanpi  8523  mulcanpi  8524  addnidpi  8525  ltexpi  8526  ltapi  8527  ltmpi  8528  indpi  8531
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-v 2790  df-dif 3155  df-in 3159  df-ss 3166  df-ni 8496
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