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Theorem 0npr 9774
Description: The empty set is not a positive real. (Contributed by NM, 15-Nov-1995.) (New usage is discouraged.)
Assertion
Ref Expression
0npr ¬ ∅ ∈ P

Proof of Theorem 0npr
StepHypRef Expression
1 eqid 2621 . 2 ∅ = ∅
2 prn0 9771 . . 3 (∅ ∈ P → ∅ ≠ ∅)
32necon2bi 2820 . 2 (∅ = ∅ → ¬ ∅ ∈ P)
41, 3ax-mp 5 1 ¬ ∅ ∈ P
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1480  wcel 1987  c0 3897  Pcnp 9641
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-v 3192  df-dif 3563  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3898  df-np 9763
This theorem is referenced by:  genpass  9791  distrpr  9810  ltaddpr2  9817  ltapr  9827  addcanpr  9828  ltsrpr  9858  ltsosr  9875  mappsrpr  9889
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