# Union is Empty iff Sets are Empty/Proof 1

$S \cup T = \O \iff S = \O \land T = \O$
 $\ds S \cup T = \varnothing$ $\iff$ $\ds \neg \exists x: x \in \left ({S \cup T}\right)$ 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
$\blacksquare$