Theorem ltrelpr 7785

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 7750	. 2 |- <P = { <. x , y >. | ( ( x e. P. /\ y e. P. ) /\ E. q e. Q. ( q e. ( 2nd ` x ) /\ q e. ( 1st ` y ) ) ) }
2		opabssxp 4806	. 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 3260	1 |- <P C_ ( P. X. P. )

Syntax hints:   /\ wa 104   e. wcel 2202   E.wrex 2512   C_ wss 3201   {copab 4154   X. cxp 4729   ` cfv 5333   1stc1st 6310   2ndc2nd 6311   Q.cnq 7560   P.cnp 7571   <P cltp 7575

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-in 3207  df-ss 3214  df-opab 4156  df-xp 4737  df-iltp 7750

This theorem is referenced by:  ltprordil  7869  ltexprlemm  7880  ltexprlemopl  7881  ltexprlemlol  7882  ltexprlemopu  7883  ltexprlemupu  7884  ltexprlemdisj  7886  ltexprlemloc  7887  ltexprlemfl  7889  ltexprlemrl  7890  ltexprlemfu  7891  ltexprlemru  7892  ltexpri  7893  lteupri  7897  ltaprlem  7898  prplnqu  7900  caucvgprprlemk  7963  caucvgprprlemnkltj  7969  caucvgprprlemnkeqj  7970  caucvgprprlemnjltk  7971  caucvgprprlemnbj  7973  caucvgprprlemml  7974  caucvgprprlemlol  7978  caucvgprprlemupu  7980  suplocexprlemss  7995  suplocexprlemlub  8004  gt0srpr  8028  lttrsr  8042  ltposr  8043  archsr  8062