If $x,y$ are ordinals and $x \cong y$ then $x = y$
The proof goes as follows;
Suppose $f$ is an isomorphism between $x$ and $y$ then show $f$ is the identity map.
Assume that $f$ is not the identity map. Consider the least element $z$ of $x$. s.t. $f(z) \neq z$. But then $f(w) = w,\forall w \in z$.
Now the set of predecessors of $f(z)$ is $f(w)$ for $w \in z$, which is $z$ itself. So $f(z) = z$. Contradiction.
I don't really understand the proof, how can you say $w \in z$ when $z$ is already the least element and then assume $f(w) = w$ when we've basically stated in our contradiction that it cannot.
Thanks in advance!