[FOM] A little trouble with definition of "binary relation" in Wikipedia

[Victor Makarov]

The following definition of "binary relation" one can find in Wikipedia:

( http://en.wikipedia.org/wiki/Binary_relation#Formal_definition  )

"A binary relation R is usually defined as an ordered triple (X, Y, G) where 
X and Y are arbitrary sets (or classes), and G is a subset of the Cartesian 
product X × Y."

But usually in set theories with classes (for example, NBG) an ordered 
triple (X, Y, G) is definend as

the ordered pair (X, (Y, G)); and an ordered pair(a,b)  is defined as the 
set { {a},  {a, b} }.

Because elements of sets must be sets, X, Y, G must be also sets (not proper 
classes).


My question is:

Are there set theories with classes, where proper classes can be elements of 
other classes?

Thanks in advance,

Victor Makarov