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.