TITLE: Axiom of extensionality in ZF - pointless?
QUESTION [16 upvotes]: In every textbook on ZF set theory I come across the Axiom of extensionality, which basically says that if two sets have the same elements, then those two sets are equal:
$$\forall x \forall y (\forall z(z\in x \iff z\in y)\Rightarrow x=y)$$
But why is such statement made to be an axiom? It surely looks like a straightforward definition, because it only relies on the primitive notion of membership, logical connectives and quantifiers. What am I missing here?

REPLY [3 votes]: Suppose the set $S$ contains the element $a$ and no other elements.
Suppose the set $T$ contains the element $a$ and no other elements.
Should these sets be equal? If you think so, then you'll have to use extensionality to argue as such: it is the only axiom of ZFC that allows you to infer that two sets are equal.