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Theorem 0npi 9261
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 2467 . 2  |-  (/)  =  (/)
2 elni 9255 . . . 4  |-  ( (/)  e.  N.  <->  ( (/)  e.  om  /\  (/)  =/=  (/) ) )
32simprbi 464 . . 3  |-  ( (/)  e.  N.  ->  (/)  =/=  (/) )
43necon2bi 2704 . 2  |-  ( (/)  =  (/)  ->  -.  (/)  e.  N. )
51, 4ax-mp 5 1  |-  -.  (/)  e.  N.
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    = wceq 1379    e. wcel 1767    =/= wne 2662   (/)c0 3785   omcom 6685   N.cnpi 9223
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-v 3115  df-dif 3479  df-sn 4028  df-ni 9251
This theorem is referenced by:  addasspi  9274  mulasspi  9276  distrpi  9277  addcanpi  9278  mulcanpi  9279  addnidpi  9280  ltapi  9282  ltmpi  9283  ordpipq  9321
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