MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  prn0 Unicode version

Theorem prn0 8490
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 8489 . . 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 3396 . 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 2412   A.wral 2507   E.wrex 2508   _Vcvv 2725    C. wpss 3076   (/)c0 3359   class class class wbr 3917   Q.cnq 8351    <Q cltq 8357   P.cnp 8358
This theorem is referenced by:  0npr  8493  npomex  8497  genpn0  8504  prlem934  8534  ltaddpr  8535  prlem936  8548  reclem2pr  8549  suplem1pr  8553
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 1926  ax-ext 2234
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 1883  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2511  df-rex 2512  df-v 2727  df-dif 3078  df-in 3082  df-ss 3086  df-pss 3088  df-nul 3360  df-np 8482
  Copyright terms: Public domain W3C validator