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Theorem ltrelpr 9372
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 9359 . 2  |-  <P  =  { <. x ,  y
>.  |  ( (
x  e.  P.  /\  y  e.  P. )  /\  x  C.  y ) }
2 opabssxp 5072 . 2  |-  { <. x ,  y >.  |  ( ( x  e.  P.  /\  y  e.  P. )  /\  x  C.  y ) }  C_  ( P.  X.  P. )
31, 2eqsstri 3534 1  |-  <P  C_  ( P.  X.  P. )
Colors of variables: wff setvar class
Syntax hints:    /\ wa 369    e. wcel 1767    C_ wss 3476    C. wpss 3477   {copab 4504    X. cxp 4997   P.cnp 9233    <P cltp 9237
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-in 3483  df-ss 3490  df-opab 4506  df-xp 5005  df-ltp 9359
This theorem is referenced by:  ltexpri  9417  ltaprlem  9418  ltapr  9419  suplem1pr  9426  suplem2pr  9427  supexpr  9428  ltsrpr  9450  ltsosr  9467  mappsrpr  9481
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