Collection of Sets Equivalent to Set Containing Empty Set is Proper Class

Theorem
Let $S = \set \O$ be the singleton whose element is the empty set.

The collection of all sets which are equivalent to $S$ is a proper class.

Proof
Let $C$ be the class of singletons:
 * $\set {x: \exists y: x = \set y}$

By definition of cardinality, $C$ is the collection of all sets which are equivalent to $S = \set \O$.

$C$ is a set.

Define a class mapping $f: C \to V$, where $V$ is the universal class, such that $\map f {\set x} = x$.

This is a mapping on the domain $C$, as all elements of $C$ are singletons.

Take an arbitrary $x \in V$.

Then by definition of $C$:


 * $\set x \in C$

and by definition of $f$:


 * $\map f {\set x} = x$

Thus all $x \in V$ are in the image of $f$.

By assumption, $C$ is a set.

From Image of Set under Mapping is Set, $V$ is also a set.

But from Universal Class is Proper, $V$ is not a set.

From this contradiction it follows that $C$ cannot be a set.