I need to prove that for all sets A and B, if the complement of A is a subset of B then the union of A and B is equal to U, the universal set.

So far, I've come up with this:

Suppose the complement of A is a subset of B and the union of A and B isnotequal to U. Let x be part of the union of A and B. By definition of union, x is a part of A or x is a part of B. Here is where I get stuck...what should I do next? Am I on the right track?

Thanks for any help.