Here's the theorem I am trying prove
Suppose is a set, and for every family of sets
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
and the Goal is
I will assume the negation of the goal and try to find the contradiction. Hence the
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 ?