Theorem ltrelsr 7948

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 7940	. 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 4798	. 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 3257	1 |- <R C_ ( R. X. R. )

Syntax hints:   /\ wa 104   = wceq 1395   E.wex 1538   e. wcel 2200   C_ wss 3198   <.cop 3670   class class class wbr 4086   {copab 4147   X. cxp 4721  (class class class)co 6013   [cec 6695   +P. cpp 7503   <P cltp 7505   ~R cer 7506   R.cnr 7507   <R cltr 7513

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-in 3204  df-ss 3211  df-opab 4149  df-xp 4729  df-ltr 7940

This theorem is referenced by:  gt0srpr  7958  recexgt0sr  7983  addgt0sr  7985  mulgt0sr  7988  caucvgsrlemcl  7999  caucvgsrlemasr  8000  caucvgsrlemfv  8001  map2psrprg  8015  suplocsrlemb  8016  suplocsrlempr  8017  suplocsrlem  8018  suplocsr  8019  ltresr  8049  axpre-ltirr  8092  axpre-lttrn  8094