Sequences and Maps of Sets: Complements, Traces, Two-Term Sequences, the Disjointing Sequence, and Sequences in a Subset

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 as in Sequences (i), 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.

(i) Let $(A_{n})$ be a sequence in $\mathcal{P}(X)$. There is exactly one sequence $(D_{n})$ in $\mathcal{P}(X)$ with $D_{n}=X\setminus A_{n}$ for every $n\in\mathbb{N}$.

(ii) Let $\mathcal{A}$ and $Y$ be sets. There is exactly one map $g:\mathcal{A}\to\mathcal{P}(Y)$ with $g(S)=S\cap Y$ for every $S\in\mathcal{A}$. In particular, for every set $Z$ and every sequence $(B_{n})$ in $Z$, there is exactly one sequence $(F_{n})$ in $\mathcal{P}(Y)$ with $F_{n}=B_{n}\cap Y$ for every $n\in\mathbb{N}$.

(iii) Let $U$ and $V$ be subsets of $X$. There is exactly one sequence $(E_{n})$ in $\mathcal{P}(X)$ with $E_{1}=U$, $E_{2}=V$ and $E_{n}=\emptyset$ for every $n\in\mathbb{N}$ with $n\neq1$ and $n\neq2$.

(iv) Let $(A_{n})$ be a sequence in $\mathcal{P}(X)$. There is exactly one sequence $(C_{n})$ in $\mathcal{P}(X)$ with $C_{1}=X$ and $C_{n+1}=C_{n}\cap(X\setminus A_{n})$ for every $n\in\mathbb{N}$; and, for this sequence, there is exactly one sequence $(B_{n})$ in $\mathcal{P}(X)$ with $B_{n}=A_{n}\cap C_{n}$ for every $n\in\mathbb{N}$.

(v) Let $\mathcal{G}$ be a subset of $\mathcal{P}(X)$, and let $(h_{n})$ be a sequence in $\mathcal{P}(X)$ with $h_{n}\in\mathcal{G}$ for every $n\in\mathbb{N}$. Then $(h_{n})$ is a sequence in $\mathcal{G}$.