Let be non empty families of sets.
I am trying to prove that
I needed to prove something else. I had to come up with some counterexample which gave me the idea
for the above theorem. I am trying to see if I can prove it. Following is my try.
Since this is an implication , Givens are
and the Goal is
So the givens are , in logical language ,
Now, I put the goal in logical language. I am not very sure of this, please check it
since the goal is of the form , I let z be arbitrary and suppose that
So the new list of the givens is
and the goal now becomes
again , since the goal is of the form , I let t be arbitrary and suppose that
Finally the givens are
and the goal is simply
So after breaking this logically , we can now see from the givens that we let y be z and
since , we can conclude that
since z and t are arbitrary , the result holds for all values of z and t. But I am not
yet sure if this proves that
I need this for some related problem in Velleman's book "How to prove it". Author wants the
readers to explore all the logical structure of the proof. Hence all the details.