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

Theorem prn0 8615
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 8614 . . 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 959 . . 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 187 . 2  |-  ( A  e.  P.  ->  (/)  C.  A
)
4 0pss 3494 . 2  |-  ( (/)  C.  A  <->  A  =/=  (/) )
53, 4sylib 188 1  |-  ( A  e.  P.  ->  A  =/=  (/) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934   A.wal 1529    e. wcel 1686    =/= wne 2448   A.wral 2545   E.wrex 2546   _Vcvv 2790    C. wpss 3155   (/)c0 3457   class class class wbr 4025   Q.cnq 8476    <Q cltq 8482   P.cnp 8483
This theorem is referenced by:  0npr  8618  npomex  8622  genpn0  8629  prlem934  8659  ltaddpr  8660  prlem936  8673  reclem2pr  8674  suplem1pr  8678
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1531  df-nf 1534  df-sb 1632  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-ral 2550  df-rex 2551  df-v 2792  df-dif 3157  df-in 3161  df-ss 3168  df-pss 3170  df-nul 3458  df-np 8607
  Copyright terms: Public domain W3C validator