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:


 * $\forall x: \neg \paren {x \in \O}$

From False Statement implies Every Statement:


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

Hence the result by definition of subclass.