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Theorem prn0 8029
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 8028 . . 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 916 . . 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 3107 . 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 894   A.wal 1439    e. wcel 1520    =/= wne 2180   A.wral 2272   E.wrex 2273   _Vcvv 2473    C. wpss 2792   (/)c0 3070   class class class wbr 3584   Q.cnq 7890    <Q cltq 7896   P.cnp 7897
This theorem is referenced by:  0npr  8032  npomex  8036  genpn0  8043  prlem934  8073  ltaddpr  8074  prlem936  8087  reclem2pr  8088  suplem1pr  8092
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-5 1440  ax-6 1441  ax-7 1442  ax-gen 1443  ax-8 1522  ax-11 1523  ax-17 1527  ax-12o 1560  ax-10 1574  ax-9 1580  ax-4 1587  ax-16 1773  ax-ext 2044
This theorem depends on definitions:  df-bi 175  df-or 357  df-an 358  df-3an 896  df-ex 1445  df-sb 1734  df-clab 2050  df-cleq 2055  df-clel 2058  df-ne 2182  df-ral 2276  df-rex 2277  df-v 2475  df-dif 2794  df-in 2798  df-ss 2802  df-pss 2804  df-nul 3071  df-np 8021
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