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Theorem 0npr 10862
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 2738 . 2 ∅ = ∅
2 prn0 10859 . . 3 (∅ ∈ P → ∅ ≠ ∅)
32necon2bi 2973 . 2 (∅ = ∅ → ¬ ∅ ∈ P)
41, 3ax-mp 5 1 ¬ ∅ ∈ P
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1542  wcel 2107  c0 4281  Pcnp 10729
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2709
This theorem depends on definitions:  df-bi 206  df-an 398  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2816  df-ne 2943  df-ral 3064  df-rex 3073  df-v 3446  df-dif 3912  df-in 3916  df-ss 3926  df-pss 3928  df-nul 4282  df-np 10851
This theorem is referenced by:  genpass  10879  distrpr  10898  ltaddpr2  10905  ltapr  10915  addcanpr  10916  ltsrpr  10947  ltsosr  10964  mappsrpr  10978
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