Let S be a nonzero ordinal and let x be minimal in S. Suppose x is nonempty. If x contains y then S contains y, since S is transitive. With y ∈ x, x is no longer the least element in S. Therefore every nonzero ordinal S contains 0, and 0 is the least element of the entire set S.
Let S be an ordinal and let T be a member of S. The members of T belong to S, and are well ordered by membership. Let's look at transitivity. Assume y ∈ x ∈ T. Since S is transitive x is in S, and that means y is in S. The members of S are well ordered, which implies partially ordered. With y < x < T, y < T, hence y ∈ T, and T is transitive. Therefore T is an ordinal. Every member of an ordinal is another ordinal.
If S is the successor of two different ordinals x and y, we showed (earlier) that x contains y and y contains x. This violates the well ordering on the members of S. Thus the predecessor of an ordinal, if it exists, is a unique ordinal.
Next let S be the successor of T and let T be an ordinal. Show that S is a transitive set. This follows from the fact that T is transitive, and S contains all the members of T, along with T.
Next, let U be any subset of S and find its least element. If U is the set containing T, T is the least element. Otherwise U has elements from T, x is the least of these, x is also contained in T (if T is in U), and x is the least element of U. The members of S are well ordered, and S is an ordinal. in this case S is called a successor ordinal.
If S is the successor of T, one is an ordinal iff the other is an ordinal.
Let S be the successor of T. Recall that S contains T and all the members of T. If there is some other set U such that T⊆U⊆S, then U contains all the members of T, and may contain T, but nothing else. Thus U = S or U = T. There are no sets between S and T. As a corollary, there are no ordinals between an ordinal and its successor. They are similar to integers in this regard. As described in the last section, the first ordinal is 0, the next ordinal is 1, the next ordinal is 2, and so on, with no sets squeezed in between these successor ordinals.
If the ordinal S is not 0 and not a successor ordinal, then S is, by definition, a limit ordinal.
A finite ordinal is 0, or a successor ordinal that contains no limit ordinals. An infinite ordinal is an ordinal that is not finite.
A set x is finite if it can be placed in 1-1 correspondence with a finite ordinal. Thus a function, a bijection, maps the elements of x onto the ordinals 0 through n-1 inside n. We say x contains n elements. A set is infinite if it can be placed in 1-1 correspondence with an infinite ordinal.
My textbook doesn't state that S cannot contain itself, but it's worth mentioning. Without this constraint, S could contain S, 0, 1, and 2. This is transitive, and its members are well ordered, but it's not really what we want. In practice, most people view "well ordered" as < rather than ≤ - so nothing in S, including S, should contain itself.