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Theorem ltrelpr 9271
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 9258 . 2  |-  <P  =  { <. x ,  y
>.  |  ( (
x  e.  P.  /\  y  e.  P. )  /\  x  C.  y ) }
2 opabssxp 5012 . 2  |-  { <. x ,  y >.  |  ( ( x  e.  P.  /\  y  e.  P. )  /\  x  C.  y ) }  C_  ( P.  X.  P. )
31, 2eqsstri 3487 1  |-  <P  C_  ( P.  X.  P. )
Colors of variables: wff setvar class
Syntax hints:    /\ wa 369    e. wcel 1758    C_ wss 3429    C. wpss 3430   {copab 4450    X. cxp 4939   P.cnp 9130    <P cltp 9134
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-in 3436  df-ss 3443  df-opab 4452  df-xp 4947  df-ltp 9258
This theorem is referenced by:  ltexpri  9316  ltaprlem  9317  ltapr  9318  suplem1pr  9325  suplem2pr  9326  supexpr  9327  ltsrpr  9348  ltsosr  9365  mappsrpr  9379
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