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Theorem ltrelsr 9353
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 9345 . 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 5022 . 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 3497 1  |-  <R  C_  ( R.  X.  R. )
Colors of variables: wff setvar class
Syntax hints:    /\ wa 369    = wceq 1370   E.wex 1587    e. wcel 1758    C_ wss 3439   <.cop 3994   class class class wbr 4403   {copab 4460    X. cxp 4949  (class class class)co 6203   [cec 7212    +P. cpp 9143    <P cltp 9145    ~R cer 9148   R.cnr 9149    <R cltr 9155
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-in 3446  df-ss 3453  df-opab 4462  df-xp 4957  df-ltr 9345
This theorem is referenced by:  ltsrpr  9359  ltasr  9382  recexsrlem  9385  addgt0sr  9386  mulgt0sr  9387  map2psrpr  9392  supsrlem  9393  supsr  9394  ltresr  9422  axpre-lttrn  9448
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