Your notation is off, but I know what you mean. One applies to elements of a set and the other applies to subsets. This can get confusing when we talk about classes (sets of sets). Your example though consider itself. Now to say is logically equivalent to saying whenever you have then . Clearly then we have that for every set for if that were untrue I would have to find some such that . But, there is no , so the conclusion follows. Thus, clearly but since the statement isalwaysfalse.

So to get right down to it if I say I am saying that is a set that contains elements of and to say that means that is actually in .