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Theorem 0npi 8506
Description: The empty set is not a positive integer. (Contributed by NM, 26-Aug-1995.) (New usage is discouraged.)
Assertion
Ref Expression
0npi  |-  -.  (/)  e.  N.

Proof of Theorem 0npi
StepHypRef Expression
1 eqid 2283 . 2  |-  (/)  =  (/)
2 elni 8500 . . . 4  |-  ( (/)  e.  N.  <->  ( (/)  e.  om  /\  (/)  =/=  (/) ) )
32simprbi 450 . . 3  |-  ( (/)  e.  N.  ->  (/)  =/=  (/) )
43necon2bi 2492 . 2  |-  ( (/)  =  (/)  ->  -.  (/)  e.  N. )
51, 4ax-mp 8 1  |-  -.  (/)  e.  N.
Colors of variables: wff set class
Syntax hints:   -. wn 3    = wceq 1623    e. wcel 1684    =/= wne 2446   (/)c0 3455   omcom 4656   N.cnpi 8466
This theorem is referenced by:  addasspi  8519  mulasspi  8521  distrpi  8522  addcanpi  8523  mulcanpi  8524  addnidpi  8525  ltapi  8527  ltmpi  8528  ordpipq  8566
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-ne 2448  df-v 2790  df-dif 3155  df-sn 3646  df-ni 8496
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