I am a little confused by this question: Suppose , Prove that .
By definition, I know that but not . I believe only when that it can be a subset of . I am right?
In general, it is not the case that ; however, it is possible. Such sets are called transitive. Suppose . Then for every , , i.e., . This in turn means that for every , , which explains the name.
Other examples include , and other von Neumann ordinals.