Theorem ltrelsr 7739

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 7731	. 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 4702	. 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 3189	1 |- <R C_ ( R. X. R. )

Syntax hints:   /\ wa 104   = wceq 1353   E.wex 1492   e. wcel 2148   C_ wss 3131   <.cop 3597   class class class wbr 4005   {copab 4065   X. cxp 4626  (class class class)co 5877   [cec 6535   +P. cpp 7294   <P cltp 7296   ~R cer 7297   R.cnr 7298   <R cltr 7304

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-in 3137  df-ss 3144  df-opab 4067  df-xp 4634  df-ltr 7731

This theorem is referenced by:  gt0srpr  7749  recexgt0sr  7774  addgt0sr  7776  mulgt0sr  7779  caucvgsrlemcl  7790  caucvgsrlemasr  7791  caucvgsrlemfv  7792  map2psrprg  7806  suplocsrlemb  7807  suplocsrlempr  7808  suplocsrlem  7809  suplocsr  7810  ltresr  7840  axpre-ltirr  7883  axpre-lttrn  7885