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

Theorem prn0 8858
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 8857 . . 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 3657 . 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 1549    e. wcel 1725    =/= wne 2598   A.wral 2697   E.wrex 2698   _Vcvv 2948    C. wpss 3313   (/)c0 3620   class class class wbr 4204   Q.cnq 8719    <Q cltq 8725   P.cnp 8726
This theorem is referenced by:  0npr  8861  npomex  8865  genpn0  8872  prlem934  8902  ltaddpr  8903  prlem936  8916  reclem2pr  8917  suplem1pr  8921
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-v 2950  df-dif 3315  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-np 8850
  Copyright terms: Public domain W3C validator