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Theorem ltrelsr 7539
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 7531 . 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 4608 . 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 3124 1  |-  <R  C_  ( R.  X.  R. )
Colors of variables: wff set class
Syntax hints:    /\ wa 103    = wceq 1331   E.wex 1468    e. wcel 1480    C_ wss 3066   <.cop 3525   class class class wbr 3924   {copab 3983    X. cxp 4532  (class class class)co 5767   [cec 6420    +P. cpp 7094    <P cltp 7096    ~R cer 7097   R.cnr 7098    <R cltr 7104
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119
This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-in 3072  df-ss 3079  df-opab 3985  df-xp 4540  df-ltr 7531
This theorem is referenced by:  gt0srpr  7549  recexgt0sr  7574  addgt0sr  7576  mulgt0sr  7579  caucvgsrlemcl  7590  caucvgsrlemasr  7591  caucvgsrlemfv  7592  map2psrprg  7606  suplocsrlemb  7607  suplocsrlempr  7608  suplocsrlem  7609  suplocsr  7610  ltresr  7640  axpre-ltirr  7683  axpre-lttrn  7685
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