A set is determined and sometimes even defined by all its elements. If you take away all the elements of a set, you take away the set. If it is not true that you take away all the elements of a set, you do not take away the set. Thus by Biconditional Introduction: you take away all the elements of a set if and only if you take away the set. This biconditional suggests a set and all its elements are the same. There is nothing more to a set other than all its elements. The empty set actually and technically is not a set because there are no elements to determine the empty set. The empty set is not a set just as a person who runs 0 feet does not run.
While yes, what you said about the elements tied directly to the set is true, there is more to it then that. There can, but not necessarily all the time, exist a synergistic effect by bringing the elements of a set together. For example, take into factor what it means to be "alive". There is no specific element to give onto the whole of the set the idea of becoming alive, but rather it is a product of all these individual elements being tied together in the same set, that only when together, can make something alive.