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Theorem 0npr 8616
Description: The empty set is not a positive real. (Contributed by NM, 15-Nov-1995.) (New usage is discouraged.)
Assertion
Ref Expression
0npr  |-  -.  (/)  e.  P.

Proof of Theorem 0npr
StepHypRef Expression
1 eqid 2283 . 2  |-  (/)  =  (/)
2 prn0 8613 . . 3  |-  ( (/)  e.  P.  ->  (/)  =/=  (/) )
32necon2bi 2492 . 2  |-  ( (/)  =  (/)  ->  -.  (/)  e.  P. )
41, 3ax-mp 8 1  |-  -.  (/)  e.  P.
Colors of variables: wff set class
Syntax hints:   -. wn 3    = wceq 1623    e. wcel 1684   (/)c0 3455   P.cnp 8481
This theorem is referenced by:  genpass  8633  distrpr  8652  ltaddpr2  8659  ltapr  8669  addcanpr  8670  ltsrpr  8699  ltsosr  8716  mappsrpr  8730
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-3an 936  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-ral 2548  df-rex 2549  df-v 2790  df-dif 3155  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-np 8605
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