Equivalence classes and partitions

Definition. The equivalence class of $x\in A$ is $[x]=\{y\in A:y\sim x\}$.

Nonempty. $x\in[x]$ (reflexive).

Cover. Every $x$ belongs to its own class.

Disjoint. If $[x]\cap[y]\neq\emptyset$, take $z$ in both. Then $z\sim x$ and $z\sim y$, so by symmetry and transitivity $x\sim y$. Then for any $w\in[x]$, $w\sim x\sim y$, so $w\in[y]$: $[x]\subseteq[y]$. Symmetrically, $[y]\subseteq[x]$. So $[x]=[y]$.

Conclusion. The classes form a partition of $A$; the set of classes is $A/\sim$ (the quotient).