What is strictly well-ordered under $\in$ I am studying from a set of notes (otherwise quite excellent) which does not explicitly specify some definitions of "ordering."
I have tried to accumulate the various pieces from various discussions and proofs in the text, but am having a hard time putting them together.
In one place it says strictly well-orderered by $\in$ is $n\lt m$ iff $n\in m$.
In another it says a well-ordered set is totally ordered in which every non-empty set has a minimal element. And one aspect of the definition of total ordering is that inequality satisfied by "minimal" is "weak" ($\leq$) rather than "strict" ($\lt$).
Lastly, one of the two features of the definition of an ordinal is that it is  strictly well-orderered by $\in$.


M confusion comes in a proof showing that an $\alpha$ is an ordinal because "it has a minimal element and thus is well-ordered under $\in$." Whereas, the definition of an ordinal says nothing about a minimal element. Are they the same thing?
And why does it not say: strictly well-ordered by $\in$?


I know this is probably torturous reading this and I've tried to be as clear as possible. Perhaps it would be easier to suggest a reference where these aspects are delineated. I've tried wikipedia and several well-known texts.
Thanks
 A: I don't know why your notes say what they say. Often notes are written when giving a course, and it is often convenient in introductory courses to talk about non-strict orders, and then when you reach well-orders it is necessary to talk about strict orders, and the definitions become cumbersome.
How do I know all that? Because five years ago I made that same mistake.
So we say that a set $\alpha$ is an ordinal if it is transitive and strictly well-ordered by the $\in$ relation. And for two ordinals $\alpha$ and $\beta$ we write $\alpha<\beta$ if $\alpha\in\beta$.
It is necessary to require that the well-order is strict, since otherwise if we do not assume the Axiom of Regularity, it is possible that there is some $x=\{x\}$, which is then certainly a transitive set which is well-ordered by $\in$. But it is not strictly well-ordered by $\in$, since $x\in x$. And it is also necessary to assume that the set is transitive, since every singleton $x$ such that $x\neq\{x\}$ is strictly well-ordered by $\in$, whereas you want your universe to have exactly one ordinal which is a singleton.
