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Theorem ltrelsr 9448
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 9440 . 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 5064 . 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 3519 1  |-  <R  C_  ( R.  X.  R. )
Colors of variables: wff setvar class
Syntax hints:    /\ wa 369    = wceq 1383   E.wex 1599    e. wcel 1804    C_ wss 3461   <.cop 4020   class class class wbr 4437   {copab 4494    X. cxp 4987  (class class class)co 6281   [cec 7311    +P. cpp 9242    <P cltp 9244    ~R cer 9245   R.cnr 9246    <R cltr 9252
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-in 3468  df-ss 3475  df-opab 4496  df-xp 4995  df-ltr 9440
This theorem is referenced by:  ltsrpr  9457  ltasr  9480  recexsrlem  9483  addgt0sr  9484  mulgt0sr  9485  map2psrpr  9490  supsrlem  9491  supsr  9492  ltresr  9520  axpre-lttrn  9546
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