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Theorem 0npr 9693
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 2610 . 2 ∅ = ∅
2 prn0 9690 . . 3 (∅ ∈ P → ∅ ≠ ∅)
32necon2bi 2812 . 2 (∅ = ∅ → ¬ ∅ ∈ P)
41, 3ax-mp 5 1 ¬ ∅ ∈ P
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1475  wcel 1977  c0 3874  Pcnp 9560
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-v 3175  df-dif 3543  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-np 9682
This theorem is referenced by:  genpass  9710  distrpr  9729  ltaddpr2  9736  ltapr  9746  addcanpr  9747  ltsrpr  9777  ltsosr  9794  mappsrpr  9808
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