Theorem ltrelpr 7504

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 7469	. 2 |- <P = { <. x , y >. | ( ( x e. P. /\ y e. P. ) /\ E. q e. Q. ( q e. ( 2nd ` x ) /\ q e. ( 1st ` y ) ) ) }
2		opabssxp 4701	. 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 3188	1 |- <P C_ ( P. X. P. )

Syntax hints:   /\ wa 104   e. wcel 2148   E.wrex 2456   C_ wss 3130   {copab 4064   X. cxp 4625   ` cfv 5217   1stc1st 6139   2ndc2nd 6140   Q.cnq 7279   P.cnp 7290   <P cltp 7294

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 3136  df-ss 3143  df-opab 4066  df-xp 4633  df-iltp 7469

This theorem is referenced by:  ltprordil  7588  ltexprlemm  7599  ltexprlemopl  7600  ltexprlemlol  7601  ltexprlemopu  7602  ltexprlemupu  7603  ltexprlemdisj  7605  ltexprlemloc  7606  ltexprlemfl  7608  ltexprlemrl  7609  ltexprlemfu  7610  ltexprlemru  7611  ltexpri  7612  lteupri  7616  ltaprlem  7617  prplnqu  7619  caucvgprprlemk  7682  caucvgprprlemnkltj  7688  caucvgprprlemnkeqj  7689  caucvgprprlemnjltk  7690  caucvgprprlemnbj  7692  caucvgprprlemml  7693  caucvgprprlemlol  7697  caucvgprprlemupu  7699  suplocexprlemss  7714  suplocexprlemlub  7723  gt0srpr  7747  lttrsr  7761  ltposr  7762  archsr  7781