Dynkin's Pi-Lambda Theorem

Let $X$ be a set. A family $\mathcal{P}$ of subsets of $X$ is a $\pi$-system if it is nonempty and closed under finite intersections: $A,B\in\mathcal{P}$ implies $A\cap B\in\mathcal{P}$.

A family $\mathcal{L}$ of subsets of $X$ is a $\lambda$-system if: (1) $X\in\mathcal{L}$; (2) if $A,B\in\mathcal{L}$ and $A\subseteq B$, then the relative complement $B\setminus A\in\mathcal{L}$; (3) for every nondecreasing sequence $(A_m)_{m\in\mathbb{N}}$ in $\mathcal{L}$ (that is, $A_m\subseteq A_{m+1}$ for all $m$), $\bigcup_m A_m\in\mathcal{L}$.

Theorem. If $\mathcal{P}$ is a $\pi$-system, $\mathcal{L}$ is a $\lambda$-system, and $\mathcal{P}\subseteq\mathcal{L}$, then the generated $\sigma$-algebra satisfies