Full Idea
Axiom of Powers: For each set there exists a collection of sets that contains amongst its elements all the subsets of the given set.
Gist of Idea
Powers: All the subsets of a given set form their own new powerset
Source
Brian Clegg (Infinity: Quest to Think the Unthinkable [2003], Ch.15)
Book Reference
Clegg,Brian: 'Infinity' [Robinson 2003], p.206
A Reaction
Obviously this must include the whole of the base set (i.e. not just 'proper' subsets), otherwise the new set would just be a duplicate of the base set.