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Theorem 0npi 8522
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 2296 . 2  |-  (/)  =  (/)
2 elni 8516 . . . 4  |-  ( (/)  e.  N.  <->  ( (/)  e.  om  /\  (/)  =/=  (/) ) )
32simprbi 450 . . 3  |-  ( (/)  e.  N.  ->  (/)  =/=  (/) )
43necon2bi 2505 . 2  |-  ( (/)  =  (/)  ->  -.  (/)  e.  N. )
51, 4ax-mp 8 1  |-  -.  (/)  e.  N.
Colors of variables: wff set class
Syntax hints:   -. wn 3    = wceq 1632    e. wcel 1696    =/= wne 2459   (/)c0 3468   omcom 4672   N.cnpi 8482
This theorem is referenced by:  addasspi  8535  mulasspi  8537  distrpi  8538  addcanpi  8539  mulcanpi  8540  addnidpi  8541  ltapi  8543  ltmpi  8544  ordpipq  8582
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-v 2803  df-dif 3168  df-sn 3659  df-ni 8512
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