Universal Class less Set is not Transitive

Theorem
Let $V$ be a basic universe.

Let $a \in V$ be a set.

Then:
 * $V \setminus \set a$ is not a transitive class

where $\setminus$ denotes class difference.

Proof
By definition, $V$ is the class of all sets.

As $a \in V$, by definition of $V$ it follows that $a$ is a set.

Consider the power set $\powerset a$ of $a$.

From the axiom of powers:
 * $\powerset a$ is also a set

and:
 * $\powerset {\powerset a}$ is also a set.

By definition:
 * $a \in \powerset a$

and so:
 * $\set a \in \powerset {\powerset a}$

as $\powerset {\powerset a}$ is a set, it follows that:
 * $\powerset {\powerset a} \in V$

But we have by definition of class difference that $\set a \notin V$.

Hence we have an element of an element of $V$ which is not itself in $V$.

Hence, by definition, $V \setminus \set a$ is not a transitive class.