ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  ltrelsr Unicode version

Theorem ltrelsr 7751
Description: Signed real 'less than' is a relation on signed reals. (Contributed by NM, 14-Feb-1996.)
Assertion
Ref Expression
ltrelsr  |-  <R  C_  ( R.  X.  R. )

Proof of Theorem ltrelsr
Dummy variables  x  y  z  w  v  u are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-ltr 7743 . 2  |-  <R  =  { <. x ,  y
>.  |  ( (
x  e.  R.  /\  y  e.  R. )  /\  E. z E. w E. v E. u ( ( x  =  [ <. z ,  w >. ]  ~R  /\  y  =  [ <. v ,  u >. ]  ~R  )  /\  ( z  +P.  u
)  <P  ( w  +P.  v ) ) ) }
2 opabssxp 4712 . 2  |-  { <. x ,  y >.  |  ( ( x  e.  R.  /\  y  e.  R. )  /\  E. z E. w E. v E. u ( ( x  =  [ <. z ,  w >. ]  ~R  /\  y  =  [ <. v ,  u >. ]  ~R  )  /\  ( z  +P.  u
)  <P  ( w  +P.  v ) ) ) }  C_  ( R.  X.  R. )
31, 2eqsstri 3199 1  |-  <R  C_  ( R.  X.  R. )
Colors of variables: wff set class
Syntax hints:    /\ wa 104    = wceq 1363   E.wex 1502    e. wcel 2158    C_ wss 3141   <.cop 3607   class class class wbr 4015   {copab 4075    X. cxp 4636  (class class class)co 5888   [cec 6547    +P. cpp 7306    <P cltp 7308    ~R cer 7309   R.cnr 7310    <R cltr 7316
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169
This theorem depends on definitions:  df-bi 117  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-in 3147  df-ss 3154  df-opab 4077  df-xp 4644  df-ltr 7743
This theorem is referenced by:  gt0srpr  7761  recexgt0sr  7786  addgt0sr  7788  mulgt0sr  7791  caucvgsrlemcl  7802  caucvgsrlemasr  7803  caucvgsrlemfv  7804  map2psrprg  7818  suplocsrlemb  7819  suplocsrlempr  7820  suplocsrlem  7821  suplocsr  7822  ltresr  7852  axpre-ltirr  7895  axpre-lttrn  7897
  Copyright terms: Public domain W3C validator