At the Wikipedia page for the well ordering theorem
For every set X, there exists a well-ordering with domain X.
Furthermore, a well ordering is defined to be a strict total order so that each subset has a least element.
Now I have a couple of questions about this:
Does this mean that the set X itself has a least element? After all X is a subset of itself (unless the theorem is talking about strict subsets).
Some books (e.g. Manifolds and Differential Geometry, by J.M. Lee, Appendix B) defines a well ordered set to be one where there exists a partial ordering so that every subset has a least element. And then uses the well ordering theorem together with this definition.
I can easily see that the well ordering theorem implies this (since every total ordering is a partial ordering), but is there any reason why this form is stated and not the "stronger" one directly? Are they perhaps equivalent?