In Jech's Set theory book it is stated that
If $\alpha$ is an ordinal set and $\beta\in\alpha$ then $\beta$ is an ordinal set also. Proof: this follow from the definition of ordinal set.
And the definition of ordinal set says
A set is ordinal if it is transitive and well-ordered by $\in$.
However I can't see how from the definition follows the first statement. I can see that if $\beta\in\alpha$ and $\gamma\in\beta$ then $\gamma\in\alpha$ because $\alpha$ is a transitive set, that is, $\beta\in\alpha\implies\beta\subset\alpha$.
Then $\gamma\subset \alpha$. But Im unable to see why $\beta$ must be also transitive, that is, why $\gamma\subset\beta$. Can someone clarify this point? Thank you.
EDIT: ok, seeing in wikipedia I saw that there are two different (but equivalent) definitions of transitivity. My definition is
A set $x$ is transitive if $y\in x\implies y\subset x$.
The other definition state that
A set $x$ is transitive if $y\in x\land z\in y\implies z\in x$.
Then using the second definition clearly $\beta$ is transitive (I guess that this is what @i707107 tried to point in the comments). However I'm using the first definition, what makes this not obvious. Then the question reduces to show the equivalence between the two definitions of transitivity.