Union of Empty Set

Theorem
Consider the set of sets $\mathbb S$ such that $\mathbb S$ is the empty set $\varnothing$.

Then the union of $\mathbb S$ is $\varnothing$:


 * $\displaystyle \mathbb S = \varnothing \implies \bigcup \mathbb S = \varnothing$

Proof
Let $\mathbb S = \varnothing$.

Then from the definition:
 * $\displaystyle \bigcup \mathbb S = \left\{{x: \exists X \in \mathbb S: x \in X}\right\}$

from which it follows directly:
 * $\displaystyle \bigcup \mathbb S = \varnothing$

as there are no sets in $\mathbb S$.