Theorem ltrelre 7835

Description: 'Less than' is a relation on real numbers. (Contributed by NM, 22-Feb-1996.)

Assertion

Ref	Expression
ltrelre	|- <RR C_ ( RR X. RR )

Proof of Theorem ltrelre

Dummy variables x y z w are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-lt 7827	. 2 |- <RR = { <. x , y >. | ( ( x e. RR /\ y e. RR ) /\ E. z E. w ( ( x = <. z , 0R >. /\ y = <. w , 0R >. ) /\ z <R w ) ) }
2		opabssxp 4702	. 2 |- { <. x , y >. | ( ( x e. RR /\ y e. RR ) /\ E. z E. w ( ( x = <. z , 0R >. /\ y = <. w , 0R >. ) /\ z <R w ) ) } C_ ( RR X. RR )
3	1, 2	eqsstri 3189	1 |- <RR C_ ( RR X. RR )

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   0Rc0r 7300   <R cltr 7305   RRcr 7813   <RR cltrr 7818

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-lt 7827

This theorem is referenced by:  ltresr  7841