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Theorem prn0 8467
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 8466 . . 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 958 . . 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 186 . 2  |-  ( A  e.  P.  ->  (/)  C.  A
)
4 0pss 3379 . 2  |-  ( (/)  C.  A  <->  A  =/=  (/) )
53, 4sylib 187 1  |-  ( A  e.  P.  ->  A  =/=  (/) )
Colors of variables: wff set class
Syntax hints:    -> wi 5    /\ wa 357    /\ w3a 933   A.wal 1521    e. wcel 1610    =/= wne 2400   A.wral 2495   E.wrex 2496   _Vcvv 2712    C. wpss 3059   (/)c0 3342   class class class wbr 3900   Q.cnq 8328    <Q cltq 8334   P.cnp 8335
This theorem is referenced by:  0npr  8470  npomex  8474  genpn0  8481  prlem934  8511  ltaddpr  8512  prlem936  8525  reclem2pr  8526  suplem1pr  8530
This theorem was proved from axioms:  ax-1 6  ax-2 7  ax-3 8  ax-mp 9  ax-5 1522  ax-6 1523  ax-7 1524  ax-gen 1525  ax-8 1612  ax-11 1613  ax-17 1617  ax-12o 1653  ax-10 1667  ax-9 1673  ax-4 1681  ax-16 1915  ax-ext 2222
This theorem depends on definitions:  df-bi 176  df-or 358  df-an 359  df-3an 935  df-tru 1309  df-ex 1527  df-nf 1529  df-sb 1872  df-clab 2228  df-cleq 2234  df-clel 2237  df-nfc 2362  df-ne 2402  df-ral 2499  df-rex 2500  df-v 2714  df-dif 3061  df-in 3065  df-ss 3069  df-pss 3071  df-nul 3343  df-np 8459
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