Theorem ltrelpr 7506

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 7471	. 2 |- <P = { <. x , y >. | ( ( x e. P. /\ y e. P. ) /\ E. q e. Q. ( q e. ( 2nd ` x ) /\ q e. ( 1st ` y ) ) ) }
2		opabssxp 4702	. 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 3189	1 |- <P C_ ( P. X. P. )

Syntax hints:   /\ wa 104   e. wcel 2148   E.wrex 2456   C_ wss 3131   {copab 4065   X. cxp 4626   ` cfv 5218   1stc1st 6141   2ndc2nd 6142   Q.cnq 7281   P.cnp 7292   <P cltp 7296

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 3137  df-ss 3144  df-opab 4067  df-xp 4634  df-iltp 7471

This theorem is referenced by:  ltprordil  7590  ltexprlemm  7601  ltexprlemopl  7602  ltexprlemlol  7603  ltexprlemopu  7604  ltexprlemupu  7605  ltexprlemdisj  7607  ltexprlemloc  7608  ltexprlemfl  7610  ltexprlemrl  7611  ltexprlemfu  7612  ltexprlemru  7613  ltexpri  7614  lteupri  7618  ltaprlem  7619  prplnqu  7621  caucvgprprlemk  7684  caucvgprprlemnkltj  7690  caucvgprprlemnkeqj  7691  caucvgprprlemnjltk  7692  caucvgprprlemnbj  7694  caucvgprprlemml  7695  caucvgprprlemlol  7699  caucvgprprlemupu  7701  suplocexprlemss  7716  suplocexprlemlub  7725  gt0srpr  7749  lttrsr  7763  ltposr  7764  archsr  7783