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Theorem 0npr 9373
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 2443 . 2  |-  (/)  =  (/)
2 prn0 9370 . . 3  |-  ( (/)  e.  P.  ->  (/)  =/=  (/) )
32necon2bi 2680 . 2  |-  ( (/)  =  (/)  ->  -.  (/)  e.  P. )
41, 3ax-mp 5 1  |-  -.  (/)  e.  P.
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    = wceq 1383    e. wcel 1804   (/)c0 3770   P.cnp 9240
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421
This theorem depends on definitions:  df-bi 185  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-v 3097  df-dif 3464  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-np 9362
This theorem is referenced by:  genpass  9390  distrpr  9409  ltaddpr2  9416  ltapr  9426  addcanpr  9427  ltsrpr  9457  ltsosr  9474  mappsrpr  9488
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