I'm tired and completely didn't think about . I also haven't done set theory in awhile now. Good insight! You are right. The first half of the uniqueness proof is trivial since it follows from (b) and that A exists with the given properties. Given the definition you provided for uniqueness, wouldn't A be included in the ? If so, does it now follow from (a) then that:

Then the proof is complete, assuming everything else is correct.