Questions about membership relation and ordinal definition This post follows the approach of Kelley General topology.
I ask three questions about $\in$ relation and definition of the ordinal. I think they are related and can be asked in the same post. If you don't think so please feel free to add a comment and I'll create new posts.
1.- We say that a set (class) $X$ is transitive (complete, saturated) if every element of $X$ is also a (sometimes proper) subset of it. Namely,
$$X\subseteq \{x:x\subset X\}.$$
2.- We refer as the epsilon class to the class
$$\mathcal E = \{(x,y):x\in y\}.$$
Now, an ordinal is defined as a transitive class $\alpha$ such that $\mathcal E$ is trichotomic on $\alpha$.
My first question is: How can I prove that $\mathcal E$ is transitive on an ordinal $\alpha$? Because $x\in y\in z$ doesn't imply, a priori, that $x\in z$.
My second question is about the epsilon class: To define ordinals, is $\mathcal E$ valid or we should consider
$$ \mathcal E_\alpha =\{(x,y)\in \alpha\times \alpha : x\in y\} $$
(as Enderton does)?
And finally, if our theory is allowed to work with proper classes, should we add the condition be a set in the definition of ordinal? Because later we consider the class 
$$ \mathbf {Ord}=\{\alpha:\alpha\mbox{ is an ordinal}\}, $$
but if ordinals aren't sets this class doesn't exist. Moreover, Kelley proves that $\mathbf{Ord}$ is an ordinal, but then $\mathbf{Ord}$ should be an element of itself, which violates the Axiom of Regularity.
Thanks
 A: 1. Suppose $\alpha$ is an ordinal; we want to show that $\in$ is transitive on $\alpha.$ Suppose $x,y,z\in\alpha$ and $x\in y\in z;$ we have to show that $x\in z.$
Let $w=\{x,y,z\}.$ By the Axiom of Regularity, some element of $w$ is disjoint from $w;$ since neither $y$ nor $z$ is disjoint from $w,$ we must have $x\cap w=\emptyset.$ It follows that $x\ne z$ and $z\notin x;$ since $\in$ is trichotomic on $\alpha,$ the only remaining possibility is that $x\in z.$
Note that this argument depends heavily on the rarely used Axiom of Regularity. Without this axiom, we would have to define ordinals in a more complicated way, namely, as transitive sets which are well-ordered by $\in.$
2. As far as I can see, "$\mathcal E$ is trichotomic on $\alpha$" means exactly the same thing as "$\mathcal E\cap(\alpha\times\alpha)$ is trichotomic on $\alpha.$" What distinction do you see?
3. Of course $\mathbf{Ord}\notin\mathbf{Ord};$ only a set can be an element of anything, and $\mathbf{Ord}$ is not a set. For most people, an ordinal is a kind of set; although $\mathbf{Ord}$ is sort of like an ordinal, it is not an ordinal because it's not a set. In Kelley's peculiar system, $\mathbf{Ord}$ is an ordinal, is not a set, and is not an element of itself. The apparent contradiction with the definition
$$\mathbf{Ord}=\{x:x\text{ is an ordinal}\}\tag1$$
is resolved by noting that $(1)$ really means
$$\mathbf{Ord}=\{x: x\text{ is a set and }x\text{ is an ordinal}\}.$$
In my copy of Kelley (July 1957 printing), this is stated on p. 253:

For each $\beta,\ \beta\in\{\alpha:|A\}$ if and only if $\beta$ is a set and $B.$

(Here $B$ is a formula obtain from $A$ on substituting $\beta$ for $\alpha.$)
