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?