(I assume you intend for to stand for "is a subset of" as opposed to "is a proper subset of"?)

In any case, I don't understand that definition.

Is ' ' supposed to range over members of some particular set? Or over ordinals?

And why should using as an INDEX for have anything to do with as a MEMBER of ? As far as I can tell, the way you've set it up, I could use ANY from ANY index set to prove that is in just by taking to be the empty set.

More technically, you've got a free variable ' ' in your definiens that is not in your definiendum. That won't work to make a proper definition.

What book is this problem from?