What's the difference between the Axiom of Extensionality $(A1)$ and an extensional relation?
The definitions are
$(A1) \forall x,y ( x = y \leftrightarrow \forall z ( z \in x \leftrightarrow z \in y ))$
and
Let $W$ be a binary relation on a set $Y$. The relation $W$ is called extensional if $\forall x,y \in Y ( x \neq y \rightarrow \exists z \in Y (( \langle z,x \rangle \in W \land \langle z,y \rangle \notin W) \lor (\langle z,x \rangle \notin W \land \langle z,y \rangle \in W))$.
Writing the negations inside the definition above as not-negations one can restate it as
Let $W$ be a binary relation on a set $Y$. The relation $W$ is called extensional if $\forall x,y \in Y ( x = y \leftarrow \forall z \in Y ( \langle z,x \rangle \in W \leftrightarrow \langle z,y \rangle \in W) $.
Well. Of course $\rightarrow$ is always true so that extensionality of a relation seems to be the same as the axiom of exteniosnality:
Let $W$ be a binary relation on a set $Y$. The relation $W$ is called extensional if $\forall x,y \in Y ( x = y \leftrightarrow \forall z \in Y ( \langle z,x \rangle \in W \leftrightarrow \langle z,y \rangle \in W) $.
But I must be missing something since I just proved the following claim (which was an exercise in Just/Weese):
Claim: Let $N$ be a set. Then $\langle N , \overline{\in} \rangle \models (A1)$ iff $\overline{\in}$ is an extensional relation on $N$.
Here "$\overline{\in}$" is used to indicated that $N$ is a standard model and $\overline{\in}$ is the actual $\in$ on $N$. But if my musings above are right then of course this claim holds. Furthermore, a much more general claim would hold:
If $N$ is a class then $\langle N , E \rangle \models (A1)$ iff $E$ is an extensional relation on $N$.
That is, even if $N$ is a non-standard class model of set theory, the claim still holds.
Where did I go wrong? Would you point out the difference between $(A1)$ and an extensional relation to me? Many thanks for your help.