A total ordering < on a set is a well ordering iff for any non-empty subset V, there is a least element of V. That is there is vinV with for all w in V. In particular for set inclusion as <, to show V has a least element, it is clear that T= intersection of all elements of V (subsets of the original S) has the property that for all w in V. So the only thing to verify is that T is in V.