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.

$\blacksquare$

Sources

• 2010: Raymond M. Smullyan and Melvin Fitting: Set Theory and the Continuum Problem (revised ed.) ... (previous) ... (next): Chapter $2$: Some Basics of Class-Set Theory: $\S 10$ Some useful facts about transitivity: Exercise $10.1 \ \text {(b)}$