That doesn't really answer my question. You have naive set theory, Zermelo-Fraenkel set theory, von Neumann set theory, etc. In which kind of set theory are you working?

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).