Russell's Paradox/Corollary/Proof 2

Proof

 * $\exists x: \forall y: \paren {\map \RR {x, y} \iff \neg \map \RR {y, y} }$
 * $\exists x: \forall y: \paren {\map \RR {x, y} \iff \neg \map \RR {y, y} }$

By Existential Instantiation:
 * $\forall y: \paren {\map \RR {x, y} \iff \neg \map \RR {y, y} }$

By Universal Instantiation:
 * $\map \RR {x, x} \iff \neg \map \RR {x, x} $

But this contradicts Biconditional of Negated Propositions.

We thus conclude:
 * $\not \exists x: \forall y: \paren {\map \RR {x, y} \iff \neg \map \RR {y, y} }$