I am aiming to prove that: if $\alpha$ is an ordinal number and $\beta\in\alpha$, then $\beta$ is an ordinal number.
The definition of ordinal is given by:
An ordinal number is a set that is transitive and is well-ordered by the relation $\alpha<\beta\Leftrightarrow\alpha\in\beta$
where transitive is defined to be:
$x\subseteq S$ for every $x\in S$
And well-ordered is defined to be:
For every non-empty subset $A\subseteq S$, there exists an element $m\in A$ such that $\forall x\in A$, $m\leq x$
Here is a solution：
Theorem 4: Elements of ordinals are ordinals.
Proof: Elements of ordinals are subsets and so are well-ordered. Let $x\in y \in z \in\alpha $ where $\alpha$ is an ordinal. Then since $\alpha$ is transitive, $x, y, z\in\alpha$. Since $\in$ is transitive on $\alpha$, $x\in z$.
I cannot understand the method of checking of transitivity. I think in order to prove the element $z\in\alpha$ is transitive, we need to pick any element $x\in z$ and by definition conclude $x\subseteq z$ rather than assume that $x\in y\in z\in \alpha$. In fact, I cannot understand the whole part of the argument about transitivity in this proof.
Could someone please help to explain it explicity using definition? Thanks a lot.
EDIT:NOTE: Latter I realized that the answer below has not solved my question (Sorry).This version of definition of well ordering does not mention transitivity.
EDIT:Here I am asking about how to use this version of well-orderedness to prove that $\in $ is transitive. What makes ordinal special and allows $\in$ to be transitive? Could someone please use the definition above to prove it?