Union is Empty iff Sets are Empty/Proof 1

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Theorem

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


Proof

\(\ds S \cup T = \varnothing\) \(\iff\) \(\ds \neg \exists x: x \in \left ({S \cup T}\right)\) Definition of Empty Set
\(\ds \) \(\iff\) \(\ds \forall x: \neg \left ({x \in \left ({S \cup T}\right)}\right)\) De Morgan's Laws (Predicate Logic)
\(\ds \) \(\iff\) \(\ds \forall x: \neg \left ({x \in S \lor x \in T}\right)\) Definition of Set Union
\(\ds \) \(\iff\) \(\ds \forall x: x \notin S \land x \notin T\) De Morgan's Laws: Conjunction of Negations
\(\ds \) \(\iff\) \(\ds S = \varnothing \land T = \varnothing\) Definition of Empty Set

$\blacksquare$