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Theorem ltrelpr 8808
Description: Positive real 'less than' is a relation on positive reals. (Contributed by NM, 14-Feb-1996.) (New usage is discouraged.)
Assertion
Ref Expression
ltrelpr  |-  <P  C_  ( P.  X.  P. )

Proof of Theorem ltrelpr
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-ltp 8795 . 2  |-  <P  =  { <. x ,  y
>.  |  ( (
x  e.  P.  /\  y  e.  P. )  /\  x  C.  y ) }
2 opabssxp 4890 . 2  |-  { <. x ,  y >.  |  ( ( x  e.  P.  /\  y  e.  P. )  /\  x  C.  y ) }  C_  ( P.  X.  P. )
31, 2eqsstri 3321 1  |-  <P  C_  ( P.  X.  P. )
Colors of variables: wff set class
Syntax hints:    /\ wa 359    e. wcel 1717    C_ wss 3263    C. wpss 3264   {copab 4206    X. cxp 4816   P.cnp 8667    <P cltp 8671
This theorem is referenced by:  ltexpri  8853  ltaprlem  8854  ltapr  8855  suplem1pr  8862  suplem2pr  8863  supexpr  8864  ltsrpr  8885  ltsosr  8902  mappsrpr  8916
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 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-in 3270  df-ss 3277  df-opab 4208  df-xp 4824  df-ltp 8795
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