I'm trying to reconcile in my mind the two definitions of a Dynkin system generated by a collection $\mathcal{G}$ as shown on Wikiproof.
So, the first one says that $\delta(\mathcal{G})$ is the smallest d-system generated by $\mathcal{G}$. I understand that part. The alternative definition however says explicitly that any other d-system that $\mathcal{G}$ is part of must also contain $\delta(\mathcal{G})$. How come the two statements are equivalent - isn't it possible for $\delta(\mathcal{G})$ and another d-system to contain $\mathcal{G}$ without the former being a subset of the later?
What's going on is that the intersection of a non-empty family of d-systems is a d-system. Then $\delta(G)$ is the intersection of all d-systems $D$ with $G\subseteq D$ (since that $\delta(G) $ must be a d-system). This means that if $D$ is a d-system with $D\supseteq G$ then $D\supseteq \delta(G)$.