The Families of Rays of the Real Line Are Sets of Subsets of the Real Line

In the setting of Routine Facts Used Without Citation: Arithmetic, Maps, Predicativity and Indices, let $\mathcal{P}(\mathbb{R})$ be the power set of $\mathbb{R}$, and for $a\in\mathbb{R}$ let $(a,\infty)$, $[a,\infty)$, $(-\infty,a)$ and $(-\infty,a]$ be the rays of Intervals of the Real Line (iii). Let $R(a)$ stand for one fixed one of these four forms, the same form for all $a\in\mathbb{R}$.

(i) There is exactly one set $\mathcal{R}$ such that, for every set $z$, $z\in\mathcal{R}$ if and only if $z=R(a)$ for some $a\in\mathbb{R}$; it is a subset of $\mathcal{P}(\mathbb{R})$ and is written $\{R(a):a\in\mathbb{R}\}$. In particular each of $\{(a,\infty):a\in\mathbb{R}\}$, $\{[a,\infty):a\in\mathbb{R}\}$, $\{(-\infty,a):a\in\mathbb{R}\}$ and $\{(-\infty,a]:a\in\mathbb{R}\}$ is a set of subsets of $\mathbb{R}$.