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Theorem prn0 8013
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 8012 . . 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 925 . . 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 3125 . 2  |-  ( (/)  C.  A  <->  A  =/=  (/) )
53, 4sylib 186 1  |-  ( A  e.  P.  ->  A  =/=  (/) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 357    /\ w3a 903   A.wal 1450    e. wcel 1538    =/= wne 2199   A.wral 2291   E.wrex 2292   _Vcvv 2492    C. wpss 2811   (/)c0 3088   class class class wbr 3600   Q.cnq 7874    <Q cltq 7880   P.cnp 7881
This theorem is referenced by:  0npr  8016  npomex  8020  genpn0  8027  prlem934  8057  ltaddpr  8058  prlem936  8071  reclem2pr  8072  suplem1pr  8076
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-5 1451  ax-6 1452  ax-7 1453  ax-gen 1454  ax-8 1540  ax-11 1541  ax-17 1545  ax-12o 1578  ax-10 1592  ax-9 1598  ax-4 1606  ax-16 1793  ax-ext 2064
This theorem depends on definitions:  df-bi 175  df-or 358  df-an 359  df-3an 905  df-ex 1456  df-sb 1754  df-clab 2070  df-cleq 2075  df-clel 2078  df-ne 2201  df-ral 2295  df-rex 2296  df-v 2494  df-dif 2813  df-in 2817  df-ss 2821  df-pss 2823  df-nul 3089  df-np 8005
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