[THIS SECTION NEEDS WORKING but I'm just putting things where they belong in a very quick pass so that I can get this book ORGANIZED.]
For any given object, the intersection and the union of its identity set with itself is the identity set of the object, and the intersection of the identity groups of two different objects is the empty set. This last line defines, in fact, the very essence of what we mean by ``different'' just as the first two lines encapsulate what we mean by ``the same''.
Let us relate these two statements back to the Laws of Thought. In an existential -set theory, is not a set. Our human minds try to interpret it as ``the set of things that do not exist'' within the existential set Universe in question, but of course no such set exists within that set Universe (including the empty set) - it is a statement even in English. Nevertheless, we recognize the first of these two relations as the Law of Contradiction where as usual, can be any object or the ``empty object'' corresponding to the empty set drawn from within the set Universe. Rendered in English, it says that ``the intersection of any object drawn from our Universal set and not-an-set is not-a-set'' (within the existential Universal set of the theory). Nor can we form the union of any set object to a ``-thing'' that is not, in fact a set.
In this formulation, the two -statements become requirements of consistency of a set theory. Once one defines a set Universe with its implicit existential subsets, then any sort of algebraic operation or set specification that yields a result that isn't one of these subsets must be , an inconsistent result. That isn't a disaster, but it does mean that this and any subsequent operations involving that result are also - meaningless - within the specified theory.