Dynkin's Pi-Lambda Theorem

In the setting of Routine Facts Used Without Citation: Arithmetic, Maps, Predicativity and Indices, let $X$ be a set.

(i) A subset $\mathcal{F}$ of the power set $\mathcal{P}(X)$ is a $\sigma$-algebra on $X$ if and only if it is both a $\pi$-system on $X$ and a $\lambda$-system on $X$.

(ii) Let $\mathcal{E}$ be a $\pi$-system on $X$ and $\mathcal{L}$ a $\lambda$-system on $X$ with $\mathcal{E}\subseteq\mathcal{L}$. Then the $\sigma$-algebra generated by $\mathcal{E}$ satisfies $\sigma(\mathcal{E})\subseteq\mathcal{L}$.

(iii) Let $\mathcal{E}$ be a $\pi$-system on $X$. Then $\sigma(\mathcal{E})$ is the set of all $B\in\mathcal{P}(X)$ such that $B\in\mathcal{M}$ for every $\lambda$-system $\mathcal{M}$ on $X$ with $\mathcal{E}\subseteq\mathcal{M}$.