Users' Mathboxes Mathbox for Frédéric Liné < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  mlteqer Unicode version

Theorem mlteqer 25617
Description: The members of a 'less than or equal' relationship are extended reals. (Contributed by FL, 31-Jul-2009.) (Proof shortened by Mario Carneiro, 4-May-2015.)
Assertion
Ref Expression
mlteqer  |-  ( A  <_  B  ->  ( A  e.  RR*  /\  B  e.  RR* ) )

Proof of Theorem mlteqer
StepHypRef Expression
1 lerelxr 8888 . 2  |-  <_  C_  ( RR*  X.  RR* )
21brel 4737 1  |-  ( A  <_  B  ->  ( A  e.  RR*  /\  B  e.  RR* ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    e. wcel 1684   class class class wbr 4023   RR*cxr 8866    <_ cle 8868
This theorem is referenced by:  xrletr2  25618
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-xp 4695  df-le 8873
  Copyright terms: Public domain W3C validator