From what I gather a set $x$ is $transitive$ if whenever $y \in z , z \in x \rightarrow y \in x$
And one of the properties of a well-ordered set is that $x \in y , y \in z \rightarrow x \in z$
Now I gather a set $x$ is an ordinal $\Leftrightarrow x$ is transitive and well-ordered by $\in$
My question is why is it necessary to say an ordinal is transitive, isn't it already covered in the definition of well-ordering?
(Sorry I'm probably missing something very obvious)