Union is Empty iff Sets are Empty/Set of Sets

Theorem

Let $\SS$ be a set of sets.

Then:

$\ds \bigcup \SS = \O \iff \forall S \in \SS: S = \O$

$\ds $

$\ds \bigcup \SS = \O$

$\ds \leadstoandfrom \ \ $

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

$\ds $

$\ds x \in \paren {\bigcup \SS}$

Definition of Empty Set

$\ds \leadstoandfrom \ \ $

$\ds \forall x: \, $

$\ds $

$\ds \neg \paren {x \in \bigcup \SS }$

$\ds \leadstoandfrom \ \ $

$\ds \forall x: \, $

$\ds $

$\ds \neg \paren {\exists S \in \SS : x \in S}$

Definition of Set Union

$\ds \leadstoandfrom \ \ $

$\ds \forall x, \forall S \in \SS: \, $

$\ds $

$\ds x \notin S$

$\ds \leadstoandfrom \ \ $

$\ds \forall S \in \SS: \, $

$\ds $

$\ds S = \O$

Definition of Empty Set

$\blacksquare$