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Theorem prn0 9388
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 9387 . . 3
2 simpl2 1000 . . 3
31, 2sylbi 195 . 2
4 0pss 3864 . 2
53, 4sylib 196 1
Colors of variables: wff setvar class
Syntax hints:  ->wi 4  /\wa 369  /\w3a 973  A.wal 1393  e.wcel 1818  =/=wne 2652  A.wral 2807  E.wrex 2808   cvv 3109  C.wpss 3476   c0 3784   class class class wbr 4452   cnq 9251   cltq 9257   cnp 9258
This theorem is referenced by:  0npr  9391  npomex  9395  genpn0  9402  prlem934  9432  ltaddpr  9433  prlem936  9446  reclem2pr  9447  suplem1pr  9451
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435
This theorem depends on definitions:  df-bi 185  df-an 371  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-v 3111  df-dif 3478  df-in 3482  df-ss 3489  df-pss 3491  df-nul 3785  df-np 9380
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