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Theorem ltrelpr 7281
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 7246 . 2  |-  <P  =  { <. x ,  y
>.  |  ( (
x  e.  P.  /\  y  e.  P. )  /\  E. q  e.  Q.  ( q  e.  ( 2nd `  x )  /\  q  e.  ( 1st `  y ) ) ) }
2 opabssxp 4583 . 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 3099 1  |-  <P  C_  ( P.  X.  P. )
Colors of variables: wff set class
Syntax hints:    /\ wa 103    e. wcel 1465   E.wrex 2394    C_ wss 3041   {copab 3958    X. cxp 4507   ` cfv 5093   1stc1st 6004   2ndc2nd 6005   Q.cnq 7056   P.cnp 7067    <P cltp 7071
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 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-in 3047  df-ss 3054  df-opab 3960  df-xp 4515  df-iltp 7246
This theorem is referenced by:  ltprordil  7365  ltexprlemm  7376  ltexprlemopl  7377  ltexprlemlol  7378  ltexprlemopu  7379  ltexprlemupu  7380  ltexprlemdisj  7382  ltexprlemloc  7383  ltexprlemfl  7385  ltexprlemrl  7386  ltexprlemfu  7387  ltexprlemru  7388  ltexpri  7389  lteupri  7393  ltaprlem  7394  prplnqu  7396  caucvgprprlemk  7459  caucvgprprlemnkltj  7465  caucvgprprlemnkeqj  7466  caucvgprprlemnjltk  7467  caucvgprprlemnbj  7469  caucvgprprlemml  7470  caucvgprprlemlol  7474  caucvgprprlemupu  7476  suplocexprlemss  7491  suplocexprlemlub  7500  gt0srpr  7524  lttrsr  7538  ltposr  7539  archsr  7558
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