Equivalence Classes Partition the Set: Cover, Disjointness and Representatives; the Quotient Is a Set and the Canonical Projection Is a Surjection

In the setting of Class Theory NBG: the Axioms, Standing Conventions and Basic Notation, let $a$ be a set and $R$ an equivalence relation on $a$; for $u\in a$, $[u]_{R}$ is the equivalence class of $u$, and $a/R$ and $\pi_{R}$ are the quotient of $a$ by $R$ and the canonical projection.

¶ For all $u,v\in a$, $[u]_{R}=[v]_{R}$ if and only if $u\,R\,v$.

¶ For all $u,v\in a$, if $[u]_{R}\neq[v]_{R}$ then $[u]_{R}$ and $[v]_{R}$ are disjoint.

¶ The quotient $a/R$ is a set, and it is a subclass of the power set $\mathcal{P}(a)$.

¶ Every $u\in a$ is an element of its equivalence class, $u\in[u]_{R}$; the union set $\bigcup(a/R)$ equals $a$; and every element of $a/R$ has at least one element.

¶ For every $c\in a/R$ and every $u\in c$, $u\in a$ and $c=[u]_{R}$.

¶ The canonical projection $\pi_{R}$ is a function from $a$ to $a/R$ that is surjective, and its value at every $u\in a$ is $\pi_{R}(u)=[u]_{R}$.