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.

$\blacksquare$

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: Note $1$.