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Theorem prn0 8866
 Description: A positive real is not empty. (Contributed by NM, 15-May-1996.) (Revised by Mario Carneiro, 11-May-2013.) (New usage is discouraged.)
Assertion
Ref Expression
prn0

Proof of Theorem prn0
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elnpi 8865 . . 3
2 simpl2 961 . . 3
31, 2sylbi 188 . 2
4 0pss 3665 . 2
53, 4sylib 189 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 359   w3a 936  wal 1549   wcel 1725   wne 2599  wral 2705  wrex 2706  cvv 2956   wpss 3321  c0 3628   class class class wbr 4212  cnq 8727   cltq 8733  cnp 8734 This theorem is referenced by:  0npr  8869  npomex  8873  genpn0  8880  prlem934  8910  ltaddpr  8911  prlem936  8924  reclem2pr  8925  suplem1pr  8929 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-v 2958  df-dif 3323  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-np 8858
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