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Theorem prn0 8799
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
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elnpi 8798 . . 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 961 . . 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 188 . 2  |-  ( A  e.  P.  ->  (/)  C.  A
)
4 0pss 3608 . 2  |-  ( (/)  C.  A  <->  A  =/=  (/) )
53, 4sylib 189 1  |-  ( A  e.  P.  ->  A  =/=  (/) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936   A.wal 1546    e. wcel 1717    =/= wne 2550   A.wral 2649   E.wrex 2650   _Vcvv 2899    C. wpss 3264   (/)c0 3571   class class class wbr 4153   Q.cnq 8660    <Q cltq 8666   P.cnp 8667
This theorem is referenced by:  0npr  8802  npomex  8806  genpn0  8813  prlem934  8843  ltaddpr  8844  prlem936  8857  reclem2pr  8858  suplem1pr  8862
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-v 2901  df-dif 3266  df-in 3270  df-ss 3277  df-pss 3279  df-nul 3572  df-np 8791
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