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Theorem ltrelre 8973
Description: 'Less than' is a relation on real numbers. (Contributed by NM, 22-Feb-1996.) (New usage is discouraged.)
Assertion
Ref Expression
ltrelre  |-  <RR  C_  ( RR  X.  RR )

Proof of Theorem ltrelre
Dummy variables  x  y  z  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-lt 8967 . 2  |-  <RR  =  { <. x ,  y >.  |  ( ( x  e.  RR  /\  y  e.  RR )  /\  E. z E. w ( ( x  =  <. z ,  0R >.  /\  y  =  <. w ,  0R >. )  /\  z  <R  w ) ) }
2 opabssxp 4917 . 2  |-  { <. x ,  y >.  |  ( ( x  e.  RR  /\  y  e.  RR )  /\  E. z E. w ( ( x  =  <. z ,  0R >.  /\  y  =  <. w ,  0R >. )  /\  z  <R  w ) ) }  C_  ( RR  X.  RR )
31, 2eqsstri 3346 1  |-  <RR  C_  ( RR  X.  RR )
Colors of variables: wff set class
Syntax hints:    /\ wa 359   E.wex 1547    = wceq 1649    e. wcel 1721    C_ wss 3288   <.cop 3785   class class class wbr 4180   {copab 4233    X. cxp 4843   0Rc0r 8707    <R cltr 8712   RRcr 8953    <RR cltrr 8958
This theorem is referenced by:  ltresr  8979
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 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-in 3295  df-ss 3302  df-opab 4235  df-xp 4851  df-lt 8967
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