Every function f: A $\rightarrow$ P(A) is not surjective In this case the P(A) is the power set of A.
I want to prove this by contradiction, even though it's easier to say that the power set of A is a bigger infinity, I am not allowed to assume that.
So I want to proceed by contradiction. 
This is a proof by contradiction, let us assume that every function is f: A $\rightarrow$ P(A) is surjective, we shall show that this leads to a contradiction.  Consider the set S {x $\in$ A : x $\notin$ f(x)}
I am not sure where to go from here any advice? 

Since $f$ is surjective there is some $y$ such that $f(y) = S$. Then is $y \in S$?
