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Theorem ltrelpr 8622
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 8609 . 2  |-  <P  =  { <. x ,  y
>.  |  ( (
x  e.  P.  /\  y  e.  P. )  /\  x  C.  y ) }
2 opabssxp 4762 . 2  |-  { <. x ,  y >.  |  ( ( x  e.  P.  /\  y  e.  P. )  /\  x  C.  y ) }  C_  ( P.  X.  P. )
31, 2eqsstri 3208 1  |-  <P  C_  ( P.  X.  P. )
Colors of variables: wff set class
Syntax hints:    /\ wa 358    e. wcel 1684    C_ wss 3152    C. wpss 3153   {copab 4076    X. cxp 4687   P.cnp 8481    <P cltp 8485
This theorem is referenced by:  ltexpri  8667  ltaprlem  8668  ltapr  8669  suplem1pr  8676  suplem2pr  8677  supexpr  8678  ltsrpr  8699  ltsosr  8716  mappsrpr  8730
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-in 3159  df-ss 3166  df-opab 4078  df-xp 4695  df-ltp 8609
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