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 $$\varnothing$$:


 * $$\mathbb S = \left\{{\varnothing}\right\} \implies \bigcup \mathbb S = \varnothing$$

Proof
Let $$\mathbb S = \left\{{\varnothing}\right\}$$.

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

from which it follows directly:
 * $$\bigcup \mathbb S = \left\{{x: x \in \varnothing}\right\}$$

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

That is:
 * $$\bigcup \mathbb S = \varnothing$$