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Theorem 0npr 8632
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 2296 . 2  |-  (/)  =  (/)
2 prn0 8629 . . 3  |-  ( (/)  e.  P.  ->  (/)  =/=  (/) )
32necon2bi 2505 . 2  |-  ( (/)  =  (/)  ->  -.  (/)  e.  P. )
41, 3ax-mp 8 1  |-  -.  (/)  e.  P.
Colors of variables: wff set class
Syntax hints:   -. wn 3    = wceq 1632    e. wcel 1696   (/)c0 3468   P.cnp 8497
This theorem is referenced by:  genpass  8649  distrpr  8668  ltaddpr2  8675  ltapr  8685  addcanpr  8686  ltsrpr  8715  ltsosr  8732  mappsrpr  8746
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-v 2803  df-dif 3168  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-np 8621
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