Swelled Class contains Empty Set

Theorem

Let $A$ be a swelled class.

Then the empty set is an element of $A$.

Proof

By definition of swelled class, every subclass of every element of $A$ is also an element of $A$.

Let $x \in A$.

Then by Empty Class is Subclass of All Classes, the empty class is an element of $A$.

By the Axiom of the Empty Set, the empty class is a set.

Hence the result.

$\blacksquare$