Every element $z$ in $X$ is in exactly one equivalence class. Not sure how to prove this.
I proved that every element $z$ in $X$ is in some equivalence class by using the definition of $[x]$. How would I prove that it is in exactly one equivalence class?
If $x$ is in two different equivalence class then neither is empty.
For any $y$ in the first equivalence class then $y \equiv x$, and for any $z$ in the second equivalence class then $x \equiv y$. So by the transitivity of equivalence $y \equiv z$ implying that $y$ is in the second equivalence class and $z$ is in the first equivalence class. So the two equivalence classes have the same members and are not different, contradicting the assumption.
So $x$ is not in more than one equivalence class.
But it is in at least one, the equivalence class of things equivalent to $x$.