Barber Paradox/Analysis 2

Analysis
Let $\map M x$ be defined as:
 * $x$ is a man in the community.

Let $\map S {x, y}$ be defined as:
 * $x$ shaves $y$.

Let $b$ be the barber.

Suppose $\map M b$.

Suppose that:
 * $\forall x, y: \paren {\map S {x, y} \implies \map M x, \map M y}$

Suppose to the contrary that:
 * $\forall x: \paren {\map M x \implies \paren {\map S {b, x} \iff \neg \map S {x, x} } }$

For $x = b$ we obtain the contradiction:
 * $\map S {b, b} \iff \neg \map S {b, b}$

Therefore, it must be false that:
 * $\forall x: \paren {\map M x \implies \paren {\map S {b, x} \iff \neg \map S {x, x} } }$