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Theorem 0npi 8761
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 2438 . 2  |-  (/)  =  (/)
2 elni 8755 . . . 4  |-  ( (/)  e.  N.  <->  ( (/)  e.  om  /\  (/)  =/=  (/) ) )
32simprbi 452 . . 3  |-  ( (/)  e.  N.  ->  (/)  =/=  (/) )
43necon2bi 2652 . 2  |-  ( (/)  =  (/)  ->  -.  (/)  e.  N. )
51, 4ax-mp 8 1  |-  -.  (/)  e.  N.
Colors of variables: wff set class
Syntax hints:   -. wn 3    = wceq 1653    e. wcel 1726    =/= wne 2601   (/)c0 3630   omcom 4847   N.cnpi 8721
This theorem is referenced by:  addasspi  8774  mulasspi  8776  distrpi  8777  addcanpi  8778  mulcanpi  8779  addnidpi  8780  ltapi  8782  ltmpi  8783  ordpipq  8821
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-v 2960  df-dif 3325  df-sn 3822  df-ni 8751
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