# Empty Class is Subclass of All Classes

## Theorem

The empty class is a subclass of all classes.

## Proof

Let $A$ be a class.

By definition of the empty class:

$\forall x: \neg \paren {x \in \O}$
$\forall x: \paren {x \in \O \implies x \in A}$

Hence the result by definition of subclass.

$\blacksquare$