Theorem ltrelsr 7546

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 7538	. 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 4613	. 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 3129	1 |- <R C_ ( R. X. R. )

Syntax hints:   /\ wa 103   = wceq 1331   E.wex 1468   e. wcel 1480   C_ wss 3071   <.cop 3530   class class class wbr 3929   {copab 3988   X. cxp 4537  (class class class)co 5774   [cec 6427   +P. cpp 7101   <P cltp 7103   ~R cer 7104   R.cnr 7105   <R cltr 7111

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-in 3077  df-ss 3084  df-opab 3990  df-xp 4545  df-ltr 7538

This theorem is referenced by:  gt0srpr  7556  recexgt0sr  7581  addgt0sr  7583  mulgt0sr  7586  caucvgsrlemcl  7597  caucvgsrlemasr  7598  caucvgsrlemfv  7599  map2psrprg  7613  suplocsrlemb  7614  suplocsrlempr  7615  suplocsrlem  7616  suplocsr  7617  ltresr  7647  axpre-ltirr  7690  axpre-lttrn  7692