All different equivalence classes are disjoint

up:: Equivalence Class

Let $\sim$ be an Equivalence Relation.

Let $[x],[y]$ two equivalence classes, and suppose that there is some point $z\in X$ in common to both.

Then $\forall x\in X,x\sim z$ and $\forall y\in Y,y\sim z⟺z\sim y$. But, by definition, $x\sim z\wedge z\sim y⟹x\sim y$.

Thus, all elements of $x$ are equivalent to $y$, and thus $[x]=[y]$.