Two sentences from the book [1]: "A possible relation between sets, more elementary than belonging, is equality." and "Two sets are equal if and only if they ha

Two sentences from the book [1]: "A possible relation between sets, more elementary than belonging, is equality." and "Two sets are equal if and only if they have the same elements". These already seem to be contradicting each other -- how can equality be more basic than belonging if the former is defined in terms of the latter. Axiom of extensionality states that the sets A and B are equal if and only if they have the same elements. The point here is this: if A and B are the same set, then whenever some x belongs to A, it also belongs to B, because B is just a different name for A. Similarly, if something belongs to B, it also belongs to A. So, the implication "A is B implies A and B have the same elements" is just a logical tautology, axiom of extensionality does not say anything new in this case. Now suppose you have two sets, C and D, such that whenever something belongs to C, it also belongs to D, and when something belongs to D, it also belongs to C -- in other words, they have the same elements. Can we conclude that C is the same set as D? Without extensionality axiom, no. That's why we usually assume axiom of extensionality, which tells exactly that whenever sets the same elements, they are the same, so that we don't end up with a weird situation where we have two sets that are indistinguishable with set theory, yet are different.