Not Every Class is a Set/Proof 1

Proof
Let a set $x$ be defined as ordinary $x \notin x$.

Let $\map \phi x$ be the set property defined as:
 * $\map \pi x := \neg \paren {x \in x}$

Then by the axiom of specification there exists a class, which can be denoted $A$, such that:
 * $A = \set {x: \neg \paren {x \in x} }$

By the axiom of extension $A$ is unique.

Thus $A$ is the class of all ordinary sets.

Hence we have:
 * $x \in A \iff x \notin x$

$A$ were a set.

Then we could set $A$ for $x$, and so obtain:
 * $A \in A \iff A \notin A$

This is a contradiction.

Hence by Proof by Contradiction $A$ cannot be a set.

But as $A$ is a subclass of the universal class $V$, $A$ is a class.

So the class $A$ of ordinary sets is not a set.

That is:
 * $A \notin V$