The Disjointing Sequence of a Sequence of Sets

In the setting of Routine Facts Used Without Citation: Arithmetic, Maps, Predicativity and Indices, let $X$ be a set and let $\mathcal{P}(X)$ be its power set. Sequences are indexed by the set $\mathbb{N}$ of natural numbers, with the number $1$ of The Set of Natural Numbers and the Number One (ii), the sum $n+1$ of Addition on Omega (i), the order $\le$ and the strict order $<$; differences and intersections of sets are those of The Boolean Operations on Classes, Disjointness, and the Universal Class (i), and $\emptyset$ is the empty set. For a sequence $(D_{n})$ in $\mathcal{P}(X)$, $\bigcup_{n\in\mathbb{N}}D_{n}$ is the union of $(D_{n})$ as a family indexed by $\mathbb{N}$. Let $(A_{n})$ be a sequence in $\mathcal{P}(X)$, and let $(C_{n})$ and $(B_{n})$ be the sequences in $\mathcal{P}(X)$ of Sequences and Maps of Sets: Complements, Traces, Two-Term Sequences, the Disjointing Sequence, and Sequences in a Subset (iv), so that $C_{1}=X$, $C_{n+1}=C_{n}\cap(X\setminus A_{n})$ and $B_{n}=A_{n}\cap C_{n}$ for every $n\in\mathbb{N}$.

(i) For every $n\in\mathbb{N}$ and every set $x$: $x\in C_{n}$ if and only if $x\in X$ and $x\notin A_{k}$ for every $k\in\mathbb{N}$ with $k<n$; and $x\in B_{n}$ if and only if $x\in A_{n}$ and $x\notin A_{k}$ for every $k\in\mathbb{N}$ with $k<n$.

(ii) $B_{n}\subseteq A_{n}$ and $B_{n}\subseteq C_{n}$ for every $n\in\mathbb{N}$.

(iii) For all $m,k\in\mathbb{N}$ with $m\neq k$, the sets $B_{m}$ and $B_{k}$ are disjoint, and $B_{m}\cap B_{k}=\emptyset$.

(iv) For every $n\in\mathbb{N}$ and every $x\in A_{n}$ there is $m\in\mathbb{N}$ with $m\le n$ and $x\in B_{m}$. Consequently $\bigcup_{n\in\mathbb{N}}B_{n}=\bigcup_{n\in\mathbb{N}}A_{n}$.

(v) Let $\mathcal{F}$ be a $\sigma$-algebra on $X$ with $A_{n}\in\mathcal{F}$ for every $n\in\mathbb{N}$. Then $C_{n}\in\mathcal{F}$ and $B_{n}\in\mathcal{F}$ for every $n\in\mathbb{N}$, and $(B_{n})$ is a sequence in $\mathcal{F}$.