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.

$\blacksquare$

Also see

• Empty Set is Unique

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: $\S 3$ Axiom of the empty set