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Theorem ltrelpr 8867
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 8854 . 2  |-  <P  =  { <. x ,  y
>.  |  ( (
x  e.  P.  /\  y  e.  P. )  /\  x  C.  y ) }
2 opabssxp 4942 . 2  |-  { <. x ,  y >.  |  ( ( x  e.  P.  /\  y  e.  P. )  /\  x  C.  y ) }  C_  ( P.  X.  P. )
31, 2eqsstri 3370 1  |-  <P  C_  ( P.  X.  P. )
Colors of variables: wff set class
Syntax hints:    /\ wa 359    e. wcel 1725    C_ wss 3312    C. wpss 3313   {copab 4257    X. cxp 4868   P.cnp 8726    <P cltp 8730
This theorem is referenced by:  ltexpri  8912  ltaprlem  8913  ltapr  8914  suplem1pr  8921  suplem2pr  8922  supexpr  8923  ltsrpr  8944  ltsosr  8961  mappsrpr  8975
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-in 3319  df-ss 3326  df-opab 4259  df-xp 4876  df-ltp 8854
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