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Theorem ltrelpr 9376
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 9363 . 2  |-  <P  =  { <. x ,  y
>.  |  ( (
x  e.  P.  /\  y  e.  P. )  /\  x  C.  y ) }
2 opabssxp 5061 . 2  |-  { <. x ,  y >.  |  ( ( x  e.  P.  /\  y  e.  P. )  /\  x  C.  y ) }  C_  ( P.  X.  P. )
31, 2eqsstri 3517 1  |-  <P  C_  ( P.  X.  P. )
Colors of variables: wff setvar class
Syntax hints:    /\ wa 369    e. wcel 1802    C_ wss 3459    C. wpss 3460   {copab 4491    X. cxp 4984   P.cnp 9237    <P cltp 9241
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-in 3466  df-ss 3473  df-opab 4493  df-xp 4992  df-ltp 9363
This theorem is referenced by:  ltexpri  9421  ltaprlem  9422  ltapr  9423  suplem1pr  9430  suplem2pr  9431  supexpr  9432  ltsrpr  9454  ltsosr  9471  mappsrpr  9485
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