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Theorem 0npi 9307
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 2451 . 2  |-  (/)  =  (/)
2 elni 9301 . . . 4  |-  ( (/)  e.  N.  <->  ( (/)  e.  om  /\  (/)  =/=  (/) ) )
32simprbi 466 . . 3  |-  ( (/)  e.  N.  ->  (/)  =/=  (/) )
43necon2bi 2654 . 2  |-  ( (/)  =  (/)  ->  -.  (/)  e.  N. )
51, 4ax-mp 5 1  |-  -.  (/)  e.  N.
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    = wceq 1444    e. wcel 1887    =/= wne 2622   (/)c0 3731   omcom 6692   N.cnpi 9269
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431
This theorem depends on definitions:  df-bi 189  df-an 373  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-v 3047  df-dif 3407  df-sn 3969  df-ni 9297
This theorem is referenced by:  addasspi  9320  mulasspi  9322  distrpi  9323  addcanpi  9324  mulcanpi  9325  addnidpi  9326  ltapi  9328  ltmpi  9329  ordpipq  9367
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