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Theorem prn0 8826
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 8825 . . 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 3629 . 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 1721    =/= wne 2571   A.wral 2670   E.wrex 2671   _Vcvv 2920    C. wpss 3285   (/)c0 3592   class class class wbr 4176   Q.cnq 8687    <Q cltq 8693   P.cnp 8694
This theorem is referenced by:  0npr  8829  npomex  8833  genpn0  8840  prlem934  8870  ltaddpr  8871  prlem936  8884  reclem2pr  8885  suplem1pr  8889
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 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389
This theorem depends on definitions:  df-bi 178  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-ral 2675  df-rex 2676  df-v 2922  df-dif 3287  df-in 3291  df-ss 3298  df-pss 3300  df-nul 3593  df-np 8818
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