Theorem ltrelsr 8018

Description: Signed real 'less than' is a relation on signed reals. (Contributed by NM, 14-Feb-1996.)

Assertion

Ref	Expression
ltrelsr	|- <R C_ ( R. X. R. )

Proof of Theorem ltrelsr

Dummy variables x y z w v u are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-ltr 8010	. 2 |- <R = { <. x , y >. | ( ( x e. R. /\ y e. R. ) /\ E. z E. w E. v E. u ( ( x = [ <. z , w >. ] ~R /\ y = [ <. v , u >. ] ~R ) /\ ( z +P. u ) <P ( w +P. v ) ) ) }
2		opabssxp 4806	. 2 |- { <. x , y >. | ( ( x e. R. /\ y e. R. ) /\ E. z E. w E. v E. u ( ( x = [ <. z , w >. ] ~R /\ y = [ <. v , u >. ] ~R ) /\ ( z +P. u ) <P ( w +P. v ) ) ) } C_ ( R. X. R. )
3	1, 2	eqsstri 3260	1 |- <R C_ ( R. X. R. )

Syntax hints:   /\ wa 104   = wceq 1398   E.wex 1541   e. wcel 2202   C_ wss 3201   <.cop 3676   class class class wbr 4093   {copab 4154   X. cxp 4729  (class class class)co 6028   [cec 6743   +P. cpp 7573   <P cltp 7575   ~R cer 7576   R.cnr 7577   <R cltr 7583

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-ltr 8010

This theorem is referenced by:  gt0srpr  8028  recexgt0sr  8053  addgt0sr  8055  mulgt0sr  8058  caucvgsrlemcl  8069  caucvgsrlemasr  8070  caucvgsrlemfv  8071  map2psrprg  8085  suplocsrlemb  8086  suplocsrlempr  8087  suplocsrlem  8088  suplocsr  8089  ltresr  8119  axpre-ltirr  8162  axpre-lttrn  8164