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Theorem 0npr 10026
 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 2760 . 2 ∅ = ∅
2 prn0 10023 . . 3 (∅ ∈ P → ∅ ≠ ∅)
32necon2bi 2962 . 2 (∅ = ∅ → ¬ ∅ ∈ P)
41, 3ax-mp 5 1 ¬ ∅ ∈ P
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   = wceq 1632   ∈ wcel 2139  ∅c0 4058  Pcnp 9893 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-v 3342  df-dif 3718  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-np 10015 This theorem is referenced by:  genpass  10043  distrpr  10062  ltaddpr2  10069  ltapr  10079  addcanpr  10080  ltsrpr  10110  ltsosr  10127  mappsrpr  10141
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