Theorem ltrelsr 7798

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 7790	. 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 4733	. 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 3211	1 |- <R C_ ( R. X. R. )

Syntax hints:   /\ wa 104   = wceq 1364   E.wex 1503   e. wcel 2164   C_ wss 3153   <.cop 3621   class class class wbr 4029   {copab 4089   X. cxp 4657  (class class class)co 5918   [cec 6585   +P. cpp 7353   <P cltp 7355   ~R cer 7356   R.cnr 7357   <R cltr 7363

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-in 3159  df-ss 3166  df-opab 4091  df-xp 4665  df-ltr 7790

This theorem is referenced by:  gt0srpr  7808  recexgt0sr  7833  addgt0sr  7835  mulgt0sr  7838  caucvgsrlemcl  7849  caucvgsrlemasr  7850  caucvgsrlemfv  7851  map2psrprg  7865  suplocsrlemb  7866  suplocsrlempr  7867  suplocsrlem  7868  suplocsr  7869  ltresr  7899  axpre-ltirr  7942  axpre-lttrn  7944