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Theorem ltrelpr 6757
Description: Positive real 'less than' is a relation on positive reals. (Contributed by NM, 14-Feb-1996.)
Assertion
Ref Expression
ltrelpr  |-  <P  C_  ( P.  X.  P. )

Proof of Theorem ltrelpr
Dummy variables  x  q  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-iltp 6722 . 2  |-  <P  =  { <. x ,  y
>.  |  ( (
x  e.  P.  /\  y  e.  P. )  /\  E. q  e.  Q.  ( q  e.  ( 2nd `  x )  /\  q  e.  ( 1st `  y ) ) ) }
2 opabssxp 4440 . 2  |-  { <. x ,  y >.  |  ( ( x  e.  P.  /\  y  e.  P. )  /\  E. q  e.  Q.  ( q  e.  ( 2nd `  x )  /\  q  e.  ( 1st `  y ) ) ) }  C_  ( P.  X.  P. )
31, 2eqsstri 3030 1  |-  <P  C_  ( P.  X.  P. )
Colors of variables: wff set class
Syntax hints:    /\ wa 102    e. wcel 1434   E.wrex 2350    C_ wss 2974   {copab 3846    X. cxp 4369   ` cfv 4932   1stc1st 5796   2ndc2nd 5797   Q.cnq 6532   P.cnp 6543    <P cltp 6547
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-in 2980  df-ss 2987  df-opab 3848  df-xp 4377  df-iltp 6722
This theorem is referenced by:  ltprordil  6841  ltexprlemm  6852  ltexprlemopl  6853  ltexprlemlol  6854  ltexprlemopu  6855  ltexprlemupu  6856  ltexprlemdisj  6858  ltexprlemloc  6859  ltexprlemfl  6861  ltexprlemrl  6862  ltexprlemfu  6863  ltexprlemru  6864  ltexpri  6865  lteupri  6869  ltaprlem  6870  prplnqu  6872  caucvgprprlemk  6935  caucvgprprlemnkltj  6941  caucvgprprlemnkeqj  6942  caucvgprprlemnjltk  6943  caucvgprprlemnbj  6945  caucvgprprlemml  6946  caucvgprprlemlol  6950  caucvgprprlemupu  6952  gt0srpr  6987  lttrsr  7001  ltposr  7002  archsr  7020
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