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Theorem prn0 8040
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 8039 . . 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 3110 . 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 2183   A.wral 2275   E.wrex 2276   _Vcvv 2476    C. wpss 2795   (/)c0 3073   class class class wbr 3592   Q.cnq 7901    <Q cltq 7907   P.cnp 7908
This theorem is referenced by:  0npr  8043  npomex  8047  genpn0  8054  prlem934  8084  ltaddpr  8085  prlem936  8098  reclem2pr  8099  suplem1pr  8103
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 1563  ax-10 1577  ax-9 1583  ax-4 1590  ax-16 1776  ax-ext 2047
This theorem depends on definitions:  df-bi 175  df-or 357  df-an 358  df-3an 898  df-ex 1447  df-sb 1737  df-clab 2053  df-cleq 2058  df-clel 2061  df-ne 2185  df-ral 2279  df-rex 2280  df-v 2478  df-dif 2797  df-in 2801  df-ss 2805  df-pss 2807  df-nul 3074  df-np 8032
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