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Theorem ltrelpr 7482
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 7447 . 2  |-  <P  =  { <. x ,  y
>.  |  ( (
x  e.  P.  /\  y  e.  P. )  /\  E. q  e.  Q.  ( q  e.  ( 2nd `  x )  /\  q  e.  ( 1st `  y ) ) ) }
2 opabssxp 4696 . 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 3187 1  |-  <P  C_  ( P.  X.  P. )
Colors of variables: wff set class
Syntax hints:    /\ wa 104    e. wcel 2148   E.wrex 2456    C_ wss 3129   {copab 4060    X. cxp 4620   ` cfv 5211   1stc1st 6132   2ndc2nd 6133   Q.cnq 7257   P.cnp 7268    <P cltp 7272
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-in 3135  df-ss 3142  df-opab 4062  df-xp 4628  df-iltp 7447
This theorem is referenced by:  ltprordil  7566  ltexprlemm  7577  ltexprlemopl  7578  ltexprlemlol  7579  ltexprlemopu  7580  ltexprlemupu  7581  ltexprlemdisj  7583  ltexprlemloc  7584  ltexprlemfl  7586  ltexprlemrl  7587  ltexprlemfu  7588  ltexprlemru  7589  ltexpri  7590  lteupri  7594  ltaprlem  7595  prplnqu  7597  caucvgprprlemk  7660  caucvgprprlemnkltj  7666  caucvgprprlemnkeqj  7667  caucvgprprlemnjltk  7668  caucvgprprlemnbj  7670  caucvgprprlemml  7671  caucvgprprlemlol  7675  caucvgprprlemupu  7677  suplocexprlemss  7692  suplocexprlemlub  7701  gt0srpr  7725  lttrsr  7739  ltposr  7740  archsr  7759
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