Empty Class Exists and is Unique

Theorem
There is exactly one empty class.

Proof
Let $P$ be a property such that $\map P x$ is satisfied by no $x$ at all, for example:
 * $\forall x: \map P x := \neg {x = x}$

Then by the axiom of specification we can create the class $A$ such that:
 * $A := \set {x \in V \land \neg {x = x} }$

from which it is seen that $A$ has no elements.

Hence there exists an empty class.

Let $A$ and $B$ both be empty classes.

By definition, both $A$ and $B$ contain the same elements, that is, no elements at all.

By the axiom of extension, that means $A = B$.

Hence the result.

Also see

 * Empty Set is Unique