Are you doing Zermelo-Fraenkel theory?
In Z set theory (or its extensions) we will have previously proven:
~ExAy y in x
Also, we we will have previously proven that for any two sets S and T there exists the union of them (SuT) such that Ay(y in SuT <-> (y in S or y in T)).
Then, toward a contradiction, suppose Az(z in C <-> z not in A).
Then Ay y in AuC.