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Theorem ltrelsr 8881
Description: Signed real 'less than' is a relation on signed reals. (Contributed by NM, 14-Feb-1996.) (New usage is discouraged.)
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 8873 . 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 4892 . 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 3323 1  |-  <R  C_  ( R.  X.  R. )
Colors of variables: wff set class
Syntax hints:    /\ wa 359   E.wex 1547    = wceq 1649    e. wcel 1717    C_ wss 3265   <.cop 3762   class class class wbr 4155   {copab 4208    X. cxp 4818  (class class class)co 6022   [cec 6841    +P. cpp 8671    <P cltp 8673    ~R cer 8676   R.cnr 8677    <R cltr 8683
This theorem is referenced by:  ltsrpr  8887  ltasr  8910  recexsrlem  8913  addgt0sr  8914  mulgt0sr  8915  map2psrpr  8920  supsrlem  8921  supsr  8922  ltresr  8950  axpre-lttrn  8976
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-in 3272  df-ss 3279  df-opab 4210  df-xp 4826  df-ltr 8873
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