Dynkin System Closed under Disjoint Union

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

Let $D, E \in \mathcal D$ be disjoint.

Then the union $D \cup E$ is also an element of $\mathcal D$.

Proof
Define $D_1 = D, D_2 = E$, and for $n \ge 2$, $D_n = \varnothing$.

Then by Dynkin System Contains Empty Set, $D_n \in \mathcal D$ for all $n \in \N$.

Also, by Intersection with Empty Set, it follows that $\left({D_n}\right)_{n \in \N}$ is a pairwise disjoint sequence.

Hence, by property $(3)$ of a Dynkin system, have:


 * $D \cup E = \displaystyle \bigcup_{n \mathop \in \N} D_n \in \mathcal D$