# Subset of Empty Set

## Theorem

Let $A$ be a class.

Then:

$A$ is a subset of the empty set $\O$
$A$ is equal to the empty set:
$A \subseteq \O \iff A = \O$

## Proof

 $\ds A = \O$ $\leadsto$ $\ds A \subseteq \O$ Definition 2 of Set Equality
 $\ds A \subseteq \O$ $\leadsto$ $\ds A \subseteq \O \land \O \subseteq A$ Empty Set is Subset of All Sets $\ds$ $\leadsto$ $\ds A = \O$ Definition 2 of Set Equality

$\blacksquare$