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Theorem ltrelre 7370
Description: 'Less than' is a relation on real numbers. (Contributed by NM, 22-Feb-1996.)
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 7363 . 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 4512 . 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 3056 1  |-  <RR  C_  ( RR  X.  RR )
Colors of variables: wff set class
Syntax hints:    /\ wa 102    = wceq 1289   E.wex 1426    e. wcel 1438    C_ wss 2999   <.cop 3449   class class class wbr 3845   {copab 3898    X. cxp 4436   0Rc0r 6857    <R cltr 6862   RRcr 7349    <RR cltrr 7354
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-in 3005  df-ss 3012  df-opab 3900  df-xp 4444  df-lt 7363
This theorem is referenced by:  ltresr  7376
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