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Theorem prn0 8023
Description: A positive real is not empty. (Contributed by NM, 15-May-1996.) (Revised by Mario Carneiro, 11-May-2013.)
Assertion
Ref Expression
prn0  |-  ( A  e.  P.  ->  A  =/=  (/) )

Proof of Theorem prn0
StepHypRef Expression
1 elnpi 8022 . . 3  |-  ( A  e.  P.  <->  ( ( A  e.  _V  /\  (/)  C.  A  /\  A  C.  Q. )  /\  A. x  e.  A  ( A. y ( y 
<Q  x  ->  y  e.  A )  /\  E. y  e.  A  x  <Q  y ) ) )
2 simpl2 915 . . 3  |-  ( ( ( A  e.  _V  /\  (/)  C.  A  /\  A  C.  Q. )  /\  A. x  e.  A  ( A. y ( y  <Q  x  ->  y  e.  A
)  /\  E. y  e.  A  x  <Q  y ) )  ->  (/)  C.  A
)
31, 2sylbi 185 . 2  |-  ( A  e.  P.  ->  (/)  C.  A
)
4 0pss 3106 . 2  |-  ( (/)  C.  A  <->  A  =/=  (/) )
53, 4sylib 186 1  |-  ( A  e.  P.  ->  A  =/=  (/) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 356    /\ w3a 893   A.wal 1438    e. wcel 1519    =/= wne 2179   A.wral 2271   E.wrex 2272   _Vcvv 2472    C. wpss 2791   (/)c0 3069   class class class wbr 3582   Q.cnq 7884    <Q cltq 7890   P.cnp 7891
This theorem is referenced by:  0npr  8026  npomex  8030  genpn0  8037  prlem934  8067  ltaddpr  8068  prlem936  8081  reclem2pr  8082  suplem1pr  8086
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-5 1439  ax-6 1440  ax-7 1441  ax-gen 1442  ax-8 1521  ax-11 1522  ax-17 1526  ax-12o 1559  ax-10 1573  ax-9 1579  ax-4 1586  ax-16 1772  ax-ext 2043
This theorem depends on definitions:  df-bi 175  df-or 357  df-an 358  df-3an 895  df-ex 1444  df-sb 1733  df-clab 2049  df-cleq 2054  df-clel 2057  df-ne 2181  df-ral 2275  df-rex 2276  df-v 2474  df-dif 2793  df-in 2797  df-ss 2801  df-pss 2803  df-nul 3070  df-np 8015
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