I think that you're not catching my point, and recommend that you look up the word "unique" in a good dictionary, preferably of mathematics. (It's tautological

I think that you're not catching my point, and recommend that you look up the word "unique" in a good dictionary, preferably of mathematics. (It's tautological and uninformative to say that any x is the unique member of the set of things equal to x. And it's simply incorrect to say that any member of a set with multiple elements is unique wrt that set.)