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Theorem 0npr 11032
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 2737 . 2 ∅ = ∅
2 prn0 11029 . . 3 (∅ ∈ P → ∅ ≠ ∅)
32necon2bi 2971 . 2 (∅ = ∅ → ¬ ∅ ∈ P)
41, 3ax-mp 5 1 ¬ ∅ ∈ P
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1540  wcel 2108  c0 4333  Pcnp 10899
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-v 3482  df-dif 3954  df-ss 3968  df-pss 3971  df-nul 4334  df-np 11021
This theorem is referenced by:  genpass  11049  distrpr  11068  ltaddpr2  11075  ltapr  11085  addcanpr  11086  ltsrpr  11117  ltsosr  11134  mappsrpr  11148
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