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Theorem ltrelre 9014
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 9008 . 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 4953 . 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 3380 1  |-  <RR  C_  ( RR  X.  RR )
Colors of variables: wff set class
Syntax hints:    /\ wa 360   E.wex 1551    = wceq 1653    e. wcel 1726    C_ wss 3322   <.cop 3819   class class class wbr 4215   {copab 4268    X. cxp 4879   0Rc0r 8748    <R cltr 8753   RRcr 8994    <RR cltrr 8999
This theorem is referenced by:  ltresr  9020
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-in 3329  df-ss 3336  df-opab 4270  df-xp 4887  df-lt 9008
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