Question on axiom of extentionality I am following the proof of ordered pair property in the book "Set theory for guided independent study" by Goldrei. Author discusses the case $ \{x \} = \{ u \} $ and says, by ZF1 (axiom of extensionality), it follows that $ x = u $. Now axiom of extentionality is given as follows
$$ \forall x \forall y \left(x = y \longleftrightarrow \forall z(z \in x \longleftrightarrow z \in y)\right) $$
Now, I don't see the conclusion follows from ZF1. Can anybody help ?
Thanks

We are given that $\{x\} =\{u\}$.  By extensionality, this means that $\forall z (z \in \{x\} \leftrightarrow z \in \{u\})$.  Instantiating this with $z=x$ we get $x \in \{x\} \leftrightarrow x \in \{u\}$.  We do have $x \in \{x\}$, and thus $x \in \{u\}$.  By definition of $\{u\}$ this implies that $x=u$.
