>>6
I didn't think about the difference between sets and classes when writing
>>4, so you made a good point. I don't think you're entirely correct though (assuming ZFC):
For each set A one can form the set {A} with 1 (a power of 2) element. So Y contains a proper class and hence isn't a set.
Because of the axiom of pairing -
\forall x \forall y \exist z (x \in z \land y \in z) - if A is a set, so is {A}.
Additionally, Y contains only elements on which the power set operator is defined, and the power operation is defined only on sets, so Y can never contain a proper class.
Still, you're probably correct in Y being a proper class, for other reasons.
(Note that I'm not a set theorist either)