# A question on proving the uniqueness of a mathematical object

When proving that there is a unique mathematical object that satisfies a particular condition, e.g., the inverse of an element of a group, is the intuition behind it the following?

You assume that the solution set (i.e., the set of objects that satisfy this condition) contains an arbitrary number of elements and then you choose two elements arbitrarily from this set. Then, if after a set of logical steps you are able to show that these two elements are actually equal, then (by the transitive property of equality) all elements of the solution set are equal to one another and hence there is actually only one unique solution.

• I see a question mark but no question. – Matt Samuel Feb 21 '15 at 18:22
• Yes, $\ |S|\ \le 1\iff \forall x,y\in S\!:\ x = y\ \$ – Bill Dubuque Feb 21 '15 at 18:24