Union of Empty Set

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

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


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

Proof
Let $\mathbb S = \O$.

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

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

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