Confusion on understanding a proposition on equivalence classes I am given to prove this proposition on equivalence classes.

Each element of $A$ is an element of one and only one equivalent class.

The part that is confusing is one and only one. It sounds like an existential statement, the existence of an equivalence class and that that equivalence class is unique. How should one understand this proposition in "common" logic terms, namely "for every", "there exists", if..then,...?
So far here is my understanding, or say 'translation' of the proposition:

There exists a unique equivalence class such that each element of $A$ belongs to that equivalence class.

This understanding is doubtful, at least for me, because it implies that all elements of $A$ belong to one equivalence class, which is wrong. 
I am not looking for a proof, just an understanding of the proposition. Thanks!

If $E$ is an equivalence relation on $A$ then for every $a\in A$ there exists an equivalence class $B$ such that $a\in B$, and for every equivalence class $C$, $a\in C$ implies $C=B$.
So there exists such equivalence class, and it is unique.
Note, however, that $\forall a\exists B\ldots$ is not the same as $\exists B\forall a\ldots$. The former is what I wrote above, and the latter is the statement that all the elements are in the same equivalence class.
