Union is Empty iff Sets are Empty/Proof 2

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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

From Empty Set is Subset of All Sets:

$\varnothing \subseteq S$

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

Similarly for $T$.

$\blacksquare$