Equivalence relations on well-orderable sets

Question

Let $W$ be a well-orderable set. Show that $W/\equiv$ is well-orderable for any equivalence relation $\equiv$ on $W.$

Here, $W/\equiv$ is the set of all equivalence classes aka the quotient. A relation $\equiv$ is called an equivalence relation if it is reflexive, symmetric and transitive.

I want to show this within ZF set theory. I have the definition of a well-orderable set too. Please can anyone lend a hand?

Answer 1

Let $[a_1]$ and $[a_2]$ be two equivalence classes such that $a_1$ and $a_2$ are the smallest representatives of each of the equivalence classes respectively (which can be done since $W$ is well-orderable). Then define a well-ordering on the equivalence classes via:

$$[a_1] \leq [a_2] \iff a_1 \leq a_2.$$

(i.e. the equivalence classes induce a partition, and you can sort each subset in the partition via their smallest element)