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

Theorem lerelxr 9651
Description: 'Less than or equal' is a relation on extended reals. (Contributed by Mario Carneiro, 28-Apr-2015.)
Assertion
Ref Expression
lerelxr  |-  <_  C_  ( RR*  X.  RR* )

Proof of Theorem lerelxr
StepHypRef Expression
1 df-le 9635 . 2  |-  <_  =  ( ( RR*  X.  RR* )  \  `'  <  )
2 difss 3631 . 2  |-  ( (
RR*  X.  RR* )  \  `'  <  )  C_  ( RR*  X.  RR* )
31, 2eqsstri 3534 1  |-  <_  C_  ( RR*  X.  RR* )
Colors of variables: wff setvar class
Syntax hints:    \ cdif 3473    C_ wss 3476    X. cxp 4997   `'ccnv 4998   RR*cxr 9628    < clt 9629    <_ cle 9630
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-v 3115  df-dif 3479  df-in 3483  df-ss 3490  df-le 9635
This theorem is referenced by:  lerel  9652  dfle2  11354  dflt2  11355  ledm  15714  lern  15715  letsr  15717  xrsle  18249  znle  18380
  Copyright terms: Public domain W3C validator