Here's the theorem I am trying prove

Suppose is a set, and for every family of sets

we have

so prove that has exactly one element. Now there are two parts here. One is the existence part and second is uniqueness part. I have proved the

existence part using method of contradiction. I chose as a particular F to show the contradiction.

Now coming to the uniqueness part, the given is

Given ---->

and the Goal is

I will assume the negation of the goal and try to find the contradiction. Hence the

givens are

and the goal now is contradiction. Now here I am having some difficulty,

how do I choose particular ? Since

and we know that

But there could be more elements in A. Is there any way I can consider different

cases which will exhaust the possibilities ?

please comment.