Dynkin System Contains Empty Set

Theorem
Let $X$ be a set, and let $\mathcal D$ be a Dynkin system on $X$.

Then the empty set $\varnothing$ is an element of $\mathcal D$.

Proof
As $\mathcal D$ is a Dynkin system, $X \in \mathcal D$.

By Set Difference with Self is Empty Set, $X \setminus X = \varnothing$.

Hence, by property $(2)$ of a Dynkin system, $\varnothing = X \setminus X \in \mathcal D$.