Theorem ltrelpr 7064

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.

Step	Hyp	Ref	Expression
1		df-iltp 7029	. 2 |- <P = { <. x , y >. | ( ( x e. P. /\ y e. P. ) /\ E. q e. Q. ( q e. ( 2nd ` x ) /\ q e. ( 1st ` y ) ) ) }
2		opabssxp 4512	. 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. )
3	1, 2	eqsstri 3056	1 |- <P C_ ( P. X. P. )

Syntax hints:   /\ wa 102   e. wcel 1438   E.wrex 2360   C_ wss 2999   {copab 3898   X. cxp 4436   ` cfv 5015   1stc1st 5909   2ndc2nd 5910   Q.cnq 6839   P.cnp 6850   <P cltp 6854

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 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070

This theorem depends on definitions:  df-bi 115  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-in 3005  df-ss 3012  df-opab 3900  df-xp 4444  df-iltp 7029

This theorem is referenced by:  ltprordil  7148  ltexprlemm  7159  ltexprlemopl  7160  ltexprlemlol  7161  ltexprlemopu  7162  ltexprlemupu  7163  ltexprlemdisj  7165  ltexprlemloc  7166  ltexprlemfl  7168  ltexprlemrl  7169  ltexprlemfu  7170  ltexprlemru  7171  ltexpri  7172  lteupri  7176  ltaprlem  7177  prplnqu  7179  caucvgprprlemk  7242  caucvgprprlemnkltj  7248  caucvgprprlemnkeqj  7249  caucvgprprlemnjltk  7250  caucvgprprlemnbj  7252  caucvgprprlemml  7253  caucvgprprlemlol  7257  caucvgprprlemupu  7259  gt0srpr  7294  lttrsr  7308  ltposr  7309  archsr  7327