All I did was use the definition of union / intersection and complement, which is . I made a list of biconditional statements. Simply, this just means whatever is on the left side of the arrow implies whats on the right and whatever is on the right side of the arrow implies whatever is on the left side. This can be veiwed symbolically as, is equivalent to and . So, to begin the proof, just choose an arbitrary element x (since we are trying to prove the general case). From there, begin looking for ways you can sart connecting definitions using logical conectives. I usally make a list of what I am given and then write down my ultimate goal. From there start breaking down your givens and goal using definitions and start looking for ways to make connections between your givens and goal.
Easier said than done, but keep pluggin' away at it.