Hello

I've been going through a proof of Zermelo's Theorem of Well Ordering, but there's a step that I don't quite understand:

It's showing that \phi is well-ordered. Why does it follow from the fact that T is in V? From this I gather that every non-empty subset has a largest element. I can see that this definesawell-ordering, but not that the inclusion order we're working with is one. I thought maybe they meant a well-ordering exist, but at the end they construct an order on S that is preserved under inclusion in \phi, so they must mean that inclusion is the well-order.

Little help clarifying, please?