That  looks much like a “universal set” for  – it not only has as members the powerset of , and all subsets of , it also has the powerset of that powerset and so on – its members are closed under the operations of taking powerset or taking a subset.
The described set is a universal set that includes every possible subset of , including the powerset of , and all subsets of that powerset, and so on. It is closed under the operations of taking powersets or subsets.