Union is Empty iff Sets are Empty/Proof 2

Theorem

If the union of two sets is the empty set, then both are themselves empty:

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

Proof

Let $S \cup T = \O$.

We have:

\(\ds S\)

\(\subseteq\)

\(\ds S \cup T\)

Set is Subset of Union

\(\ds \leadsto \ \ \)

\(\ds S\)

\(\subseteq\)

\(\ds \O\)

by hypothesis

From Empty Set is Subset of All Sets:

$\O \subseteq S$

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

Similarly for $T$.

$\blacksquare$