Union is Empty iff Sets are Empty/Proof 1

Theorem

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

$\ds S \cup T = \varnothing$

$\iff$

$\ds \neg \exists x: x \in \left ({S \cup T}\right)$

Definition of Empty Set

$\ds $

$\iff$

$\ds \forall x: \neg \left ({x \in \left ({S \cup T}\right)}\right)$

$\ds $

$\iff$

$\ds \forall x: \neg \left ({x \in S \lor x \in T}\right)$

Definition of Set Union

$\ds $

$\iff$

$\ds \forall x: x \notin S \land x \notin T$

$\ds $

$\iff$

$\ds S = \varnothing \land T = \varnothing$

Definition of Empty Set

$\blacksquare$