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Theorem prn0 8581
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 8580 . . 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 964 . . 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 189 . 2  |-  ( A  e.  P.  ->  (/)  C.  A
)
4 0pss 3467 . 2  |-  ( (/)  C.  A  <->  A  =/=  (/) )
53, 4sylib 190 1  |-  ( A  e.  P.  ->  A  =/=  (/) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    /\ wa 360    /\ w3a 939   A.wal 1532    e. wcel 1621    =/= wne 2421   A.wral 2518   E.wrex 2519   _Vcvv 2763    C. wpss 3128   (/)c0 3430   class class class wbr 3997   Q.cnq 8442    <Q cltq 8448   P.cnp 8449
This theorem is referenced by:  0npr  8584  npomex  8588  genpn0  8595  prlem934  8625  ltaddpr  8626  prlem936  8639  reclem2pr  8640  suplem1pr  8644
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-ral 2523  df-rex 2524  df-v 2765  df-dif 3130  df-in 3134  df-ss 3141  df-pss 3143  df-nul 3431  df-np 8573
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