λ-System

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

(i) A $\lambda$-system on $X$ is a subset $\mathcal{L}$ of the power set $\mathcal{P}(X)$ such that $X\in\mathcal{L}$; the difference $X\setminus A$ lies in $\mathcal{L}$ for every $A\in\mathcal{L}$; and for every sequence $(A_{n})_{n\in\mathbb{N}}$ in $\mathcal{L}$ whose terms are pairwise disjoint, that is, $A_{m}$ and $A_{n}$ are disjoint for all $m,n\in\mathbb{N}$ with $m\neq n$, the union $\bigcup_{n\in\mathbb{N}}A_{n}$, a set by The Union and Product of a Family of Sets Indexed by a Set, and the Intersection of a Family with an Inhabited Index Class, Are Sets (i), lies in $\mathcal{L}$.