Theorem ltrelsr 9889

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

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 9881	. 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 5193	. 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 3635	1 |- <R C_ ( R. X. R. )

Syntax hints:   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   C_ wss 3574   <.cop 4183   class class class wbr 4653   {copab 4712   X. cxp 5112  (class class class)co 6650   [cec 7740   +P. cpp 9683   <P cltp 9685   ~R cer 9686   R.cnr 9687   <R cltr 9693

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-in 3581  df-ss 3588  df-opab 4713  df-xp 5120  df-ltr 9881

This theorem is referenced by:  ltsrpr  9898  ltasr  9921  recexsrlem  9924  addgt0sr  9925  mulgt0sr  9926  map2psrpr  9931  supsrlem  9932  supsr  9933  ltresr  9961  axpre-lttrn  9987