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Theorem 0npr 8871
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 2438 . 2  |-  (/)  =  (/)
2 prn0 8868 . . 3  |-  ( (/)  e.  P.  ->  (/)  =/=  (/) )
32necon2bi 2652 . 2  |-  ( (/)  =  (/)  ->  -.  (/)  e.  P. )
41, 3ax-mp 8 1  |-  -.  (/)  e.  P.
Colors of variables: wff set class
Syntax hints:   -. wn 3    = wceq 1653    e. wcel 1726   (/)c0 3630   P.cnp 8736
This theorem is referenced by:  genpass  8888  distrpr  8907  ltaddpr2  8914  ltapr  8924  addcanpr  8925  ltsrpr  8954  ltsosr  8971  mappsrpr  8985
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-v 2960  df-dif 3325  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-np 8860
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