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Theorem prn0 8078
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  |-  ( A  e.  P.  ->  A  =/=  (/) )

Proof of Theorem prn0
StepHypRef Expression
1 elnpi 8077 . . 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 924 . . 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 3147 . 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 900   A.wal 1451    e. wcel 1533    =/= wne 2218   A.wral 2311   E.wrex 2312   _Vcvv 2512    C. wpss 2831   (/)c0 3110   class class class wbr 3630   Q.cnq 7939    <Q cltq 7945   P.cnp 7946
This theorem is referenced by:  0npr  8081  npomex  8085  genpn0  8092  prlem934  8122  ltaddpr  8123  prlem936  8136  reclem2pr  8137  suplem1pr  8141
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-5 1452  ax-6 1453  ax-7 1454  ax-gen 1455  ax-8 1535  ax-11 1536  ax-17 1540  ax-12o 1574  ax-10 1588  ax-9 1594  ax-4 1601  ax-16 1787  ax-ext 2082
This theorem depends on definitions:  df-bi 175  df-or 357  df-an 358  df-3an 902  df-ex 1457  df-sb 1748  df-clab 2088  df-cleq 2093  df-clel 2096  df-ne 2220  df-ral 2315  df-rex 2316  df-v 2514  df-dif 2833  df-in 2837  df-ss 2841  df-pss 2843  df-nul 3111  df-np 8070
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