From the link it seems you have proved the result. What about the following what seems to be a counterexample?

(the set of integers), , the set of all infinite sets of integers

Now there is no set that would be minimal in ...

## A counterexample?

From the link it seems you have proved the result. What about the following what seems to be a counterexample?

(the set of integers), , the set of all infinite sets of integers

Now there is no set that would be minimal in ...