Union is Empty iff Sets are Empty/Proof 1

Theorem

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

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

$\ds $

$\ds S \cup T = \O$

$\ds \leadstoandfrom \ \ $

$\ds \neg \exists x: \, $

$\ds $

$\ds x \in \paren {S \cup T}$

Definition of Empty Set

$\ds \leadstoandfrom \ \ $

$\ds \forall x: \, $

$\ds $

$\ds \neg \paren {x \in \paren {S \cup T} }$

$\ds \leadstoandfrom \ \ $

$\ds \forall x: \, $

$\ds $

$\ds \neg \paren {x \in S \lor x \in T}$

Definition of Set Union

$\ds \leadstoandfrom \ \ $

$\ds \forall x: \, $

$\ds $

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

$\ds \leadstoandfrom \ \ $

$\ds $

$\ds S = \O \land T = \O$

Definition of Empty Set

$\blacksquare$