Statement
Slogan version
$$\sigma = \pi + \lambda$$
$\sigma$ $\pi$ $\lambda$ “sum” “product” “limit” universe nonempty universe complement complement countable union finite intersection disjoint countable union A $\sigma$algebra is a $\pi$system and a $\lambda$system, and vice versa.
Wiki version
A $\lambda$system is a synonym of a Dykin system.
$$\mathcal{P} \subseteq \mathcal{D} \Longrightarrow \sigma(\mathcal{P}) \subseteq \mathcal{D}$$
Given a $\pi$system contained in a $\lambda$system. Then the $\sigma$algebra generated by the $\pi$system is also contained in the $\lambda$system.
Wiki provided two equivalent definitions of a $\lambda$system. Both of them involve a sequence of sets $(A_n)_n$ in $\mathcal{D}$. The differences are:
 $(A_n)_n$ is incresing
 $A_n$'s are pairwise disjoint
Their target is $\bigcup_n A_n \in \mathcal{D}$. Actually, the trick in #3 below shows that these two definitions mean the same thing.
Equivalence
$(S) \Rightarrow (W)$
Given $P \in \mathcal{P} \subseteq \mathcal{D}$.

universe: $\Omega \in \mathcal{D}$

stability under complement: $P^\complement \in \mathcal{D}$

stability under countable union:
I find #3 a bit tricky.
$(W) \Rightarrow (S)$
A $\sigma$algebra is always a $\pi$system and a $\lambda$system.
Given a $\pi$system $\mathcal{D}$ that is also a $\lambda$system. Apply $(W)$ taking $\mathcal{P} = \mathcal{D}$. We arrive at $\sigma(\mathcal{D}) = \mathcal{D}$.
Examples
 $\pi$system:
 nested intervals: $\mathcal{\mathcal{P}}_1 = \lbrace (\infty,x) \mid x \in \R \rbrace$
 singletons: $\mathcal{\mathcal{P}}_2 = \lbrace \lbrace x \rbrace \in \Omega \rbrace \cup \lbrace \varnothing \rbrace$
 rectangles: $\mathcal{\mathcal{P}}_3 = \lbrace \mathcal{P}_1 \times \mathcal{P}_2 \mid \mathcal{P}_1,\mathcal{P}_2 \in \mathcal{\mathcal{P}}(\Omega) \rbrace$
 $\lambda$system:
 to be continued…