For any set $A$, there is some $x$ not in $A$. I am working through a set theory text and am having trouble giving a formal proof of the above statement. So far I have:
Suppose the contrary. Then there exists a set $A$ such that $x\in A$ for all $x$. Thus $A$ is the set of all sets, which gives rise to Russell's paradox.
Is there a more concise or direct proof of this fact using the axioms of $\sf ZFC$? I am concerned about asserting that $A$ is the "set of all sets."  

We can prove that for any set $A$, there exist some $x$ such that $x$ is not in $A$.
For any set $A$ we take $x = A$ by the axiom of foundation $A$ is not in $A$.
Hence for any set $A$, there is some $x$ not in $A$ By counterexample.
