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Theorem ltrelpr 7320
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 7285 . 2  |-  <P  =  { <. x ,  y
>.  |  ( (
x  e.  P.  /\  y  e.  P. )  /\  E. q  e.  Q.  ( q  e.  ( 2nd `  x )  /\  q  e.  ( 1st `  y ) ) ) }
2 opabssxp 4613 . 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 3129 1  |-  <P  C_  ( P.  X.  P. )
Colors of variables: wff set class
Syntax hints:    /\ wa 103    e. wcel 1480   E.wrex 2417    C_ wss 3071   {copab 3988    X. cxp 4537   ` cfv 5123   1stc1st 6036   2ndc2nd 6037   Q.cnq 7095   P.cnp 7106    <P cltp 7110
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-in 3077  df-ss 3084  df-opab 3990  df-xp 4545  df-iltp 7285
This theorem is referenced by:  ltprordil  7404  ltexprlemm  7415  ltexprlemopl  7416  ltexprlemlol  7417  ltexprlemopu  7418  ltexprlemupu  7419  ltexprlemdisj  7421  ltexprlemloc  7422  ltexprlemfl  7424  ltexprlemrl  7425  ltexprlemfu  7426  ltexprlemru  7427  ltexpri  7428  lteupri  7432  ltaprlem  7433  prplnqu  7435  caucvgprprlemk  7498  caucvgprprlemnkltj  7504  caucvgprprlemnkeqj  7505  caucvgprprlemnjltk  7506  caucvgprprlemnbj  7508  caucvgprprlemml  7509  caucvgprprlemlol  7513  caucvgprprlemupu  7515  suplocexprlemss  7530  suplocexprlemlub  7539  gt0srpr  7563  lttrsr  7577  ltposr  7578  archsr  7597
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