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Theorem 0npr 10403
 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 2822 . 2 ∅ = ∅
2 prn0 10400 . . 3 (∅ ∈ P → ∅ ≠ ∅)
32necon2bi 3041 . 2 (∅ = ∅ → ¬ ∅ ∈ P)
41, 3ax-mp 5 1 ¬ ∅ ∈ P
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   = wceq 1538   ∈ wcel 2114  ∅c0 4265  Pcnp 10270 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-ext 2794 This theorem depends on definitions:  df-bi 210  df-an 400  df-3an 1086  df-ex 1782  df-sb 2070  df-clab 2801  df-cleq 2815  df-clel 2894  df-ne 3012  df-ral 3135  df-rex 3136  df-v 3471  df-dif 3911  df-in 3915  df-ss 3925  df-pss 3927  df-nul 4266  df-np 10392 This theorem is referenced by:  genpass  10420  distrpr  10439  ltaddpr2  10446  ltapr  10456  addcanpr  10457  ltsrpr  10488  ltsosr  10505  mappsrpr  10519
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