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Theorem prn0 9270
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 9269 . . 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 992 . . 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 195 . 2  |-  ( A  e.  P.  ->  (/)  C.  A
)
4 0pss 3825 . 2  |-  ( (/)  C.  A  <->  A  =/=  (/) )
53, 4sylib 196 1  |-  ( A  e.  P.  ->  A  =/=  (/) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965   A.wal 1368    e. wcel 1758    =/= wne 2648   A.wral 2799   E.wrex 2800   _Vcvv 3078    C. wpss 3438   (/)c0 3746   class class class wbr 4401   Q.cnq 9131    <Q cltq 9137   P.cnp 9138
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-v 3080  df-dif 3440  df-in 3444  df-ss 3451  df-pss 3453  df-nul 3747  df-np 9262
This theorem is referenced by:  0npr  9273  npomex  9277  genpn0  9284  prlem934  9314  ltaddpr  9315  prlem936  9328  reclem2pr  9329  suplem1pr  9333
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