Introduction to Set Theory(Ordinal Numbers) Lemma: If $\alpha$ is an ordinal number, then $\alpha \not \in \alpha$
Proof: If $\alpha \in \alpha$, then the linearly ordered set $(\alpha, \in_{\alpha})$ has an element $x=\alpha$ such that $x \in x$, contrary to asymmetry of $\in_{\alpha}$.
The proof above is given in the book 'Introduction to Set Theory' by Hrbacek and Jech. I don't understand the part 'asymmetry of $\in_{\alpha}$'. Can anyone explain it to me?

You can look up the word "asymmetric" in the index of the book and you'll find that it is defined on page 33 (in my edition).  The asymmetry is part of $\in_\alpha$ being a linear ordering.  What they're actually using here is that $\in_\alpha$ is irreflexive although that word is maybe not used in the book.  See for example here.
