# Union is Empty iff Sets are Empty/Proof 2

## Theorem

$S \cup T = \O \iff S = \O \land T = \O$

## Proof

Let $S \cup T = \varnothing$.

We have:

 $\ds S$ $\subseteq$ $\ds S \cup T$ Set is Subset of Union $\ds \implies \ \$ $\ds S$ $\subseteq$ $\ds \varnothing$ by hypothesis
$\varnothing \subseteq S$

So it follows by definition of set equality that $S = \varnothing$.

Similarly for $T$.

$\blacksquare$