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Theorem prn0 8034
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 8033 . . 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 918 . . 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 3109 . 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 896   A.wal 1441    e. wcel 1522    =/= wne 2182   A.wral 2274   E.wrex 2275   _Vcvv 2475    C. wpss 2794   (/)c0 3072   class class class wbr 3586   Q.cnq 7895    <Q cltq 7901   P.cnp 7902
This theorem is referenced by:  0npr  8037  npomex  8041  genpn0  8048  prlem934  8078  ltaddpr  8079  prlem936  8092  reclem2pr  8093  suplem1pr  8097
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-5 1442  ax-6 1443  ax-7 1444  ax-gen 1445  ax-8 1524  ax-11 1525  ax-17 1529  ax-12o 1562  ax-10 1576  ax-9 1582  ax-4 1589  ax-16 1775  ax-ext 2046
This theorem depends on definitions:  df-bi 175  df-or 357  df-an 358  df-3an 898  df-ex 1447  df-sb 1736  df-clab 2052  df-cleq 2057  df-clel 2060  df-ne 2184  df-ral 2278  df-rex 2279  df-v 2477  df-dif 2796  df-in 2800  df-ss 2804  df-pss 2806  df-nul 3073  df-np 8026
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