Russell's Paradox/Corollary/Proof 1

Proof
there does exist such an $x$.

Let $\RR$ be such that $\map \RR {x, x}$.

Then $\neg \map \RR {x, x}$.

Hence it cannot be the case that $\map \RR {x, x}$.

Now suppose that $\neg \map \RR {x, x}$.

Then by definition of $x$ it follows that $\map \RR {x, x}$.

In both cases a contradiction results.

Hence there can be no such $x$.