Equivalence classes form a partition

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

Cover. Every $a$ is in $[a]$.

Disjoint. Suppose $[a]\cap[b]\neq\emptyset$, say $c\in[a]\cap[b]$. Then $c\sim a$ and $c\sim b$. By symmetry, $a\sim c$; by transitivity $a\sim b$. For any $x\in[a]$: $x\sim a\sim b$, so $x\in[b]$: $[a]\subseteq[b]$. Symmetrically $[b]\subseteq[a]$. Equal.

Conclusion. The classes are disjoint, nonempty, cover $A$: a partition.