What does it mean for a binary relation to be an order on "equivalence classes" under another binary relation

Question

I am a bit confused about this statement in "Introduction to Set Theory" by Hrbackek and Jech. The statement is as follows: "|A| <= |B| behaves like an ordering on the "equvivalence classes" under equvipotence". Equipotence is defined as a set A and B are equipotent iff there is a one-one function f with domain A and range B and is denoted by |A|=|B|

I would like to understand what does it mean for a binary relation R1 to be an ordering on A/R2 where R2 is a an equivalence relation in A and A/R2 is the set of all equivalence classes modulo R2.

Thanks

Answer 1

This means that if you consider the collection of equivalence classes, and this order defined $\leq$, then it is a partial order. Namely, reflexive, antisymmetric and transitive.

The problem is that usually, $|A|$ is not a set, and the collection of different equivalence classes is too big to be a set as well (even if we managed to choose a unique representative from each class). So it's not really a partial order. It's not even a partial order whose domain is a proper class, because as I wrote, the objects ordered by this order are themselves not sets but proper classes.