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Theorem ltrelre 7641
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 7633 . 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 4613 . 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 3129 1  |-  <RR  C_  ( RR  X.  RR )
Colors of variables: wff set class
Syntax hints:    /\ wa 103    = wceq 1331   E.wex 1468    e. wcel 1480    C_ wss 3071   <.cop 3530   class class class wbr 3929   {copab 3988    X. cxp 4537   0Rc0r 7106    <R cltr 7111   RRcr 7619    <RR cltrr 7624
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-in 3077  df-ss 3084  df-opab 3990  df-xp 4545  df-lt 7633
This theorem is referenced by:  ltresr  7647
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