Theorem ltrelpr 7653

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 7618	. 2 |- <P = { <. x , y >. | ( ( x e. P. /\ y e. P. ) /\ E. q e. Q. ( q e. ( 2nd ` x ) /\ q e. ( 1st ` y ) ) ) }
2		opabssxp 4767	. 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 3233	1 |- <P C_ ( P. X. P. )

Syntax hints:   /\ wa 104   e. wcel 2178   E.wrex 2487   C_ wss 3174   {copab 4120   X. cxp 4691   ` cfv 5290   1stc1st 6247   2ndc2nd 6248   Q.cnq 7428   P.cnp 7439   <P cltp 7443

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-in 3180  df-ss 3187  df-opab 4122  df-xp 4699  df-iltp 7618

This theorem is referenced by:  ltprordil  7737  ltexprlemm  7748  ltexprlemopl  7749  ltexprlemlol  7750  ltexprlemopu  7751  ltexprlemupu  7752  ltexprlemdisj  7754  ltexprlemloc  7755  ltexprlemfl  7757  ltexprlemrl  7758  ltexprlemfu  7759  ltexprlemru  7760  ltexpri  7761  lteupri  7765  ltaprlem  7766  prplnqu  7768  caucvgprprlemk  7831  caucvgprprlemnkltj  7837  caucvgprprlemnkeqj  7838  caucvgprprlemnjltk  7839  caucvgprprlemnbj  7841  caucvgprprlemml  7842  caucvgprprlemlol  7846  caucvgprprlemupu  7848  suplocexprlemss  7863  suplocexprlemlub  7872  gt0srpr  7896  lttrsr  7910  ltposr  7911  archsr  7930