proof involving power sets, and intersection

Power set of (A intersect B)=Power set of A intersect the power set of B

x is an element of D which is an element of the power set of (A intersect B). a subset of A and B must contain elements that are in both A and B so x must be an element of A and B. Since x is an element of A it must be contained in the power set of A and since x is in B it must also be in the power set of B

i think this wrong but im not sure how to fix it

An element of a Set a is an element of a subset of the powerset

Note the given wiki proof more carefully. Although x is an element of A intersection B, it is not an **element **of the power set of either nor their intersection. It is an element of a subset of the power set, yes, but not an element of the power set itself. A power set is a set containing all possible **subsets** of a given set. This really is confusing though, I know. The same concept can be applied in reverse in that although x is an element of the power set of a set, it is not an element of the set itself.