Union is Empty iff Sets are Empty/Proof 2
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- $S \cup T = \O \iff S = \O \land T = \O$
Let $S \cup T = \varnothing$.
|\(\ds S\)||\(\subseteq\)||\(\ds S \cup T\)||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 $T$.