A collection  of subsets of a set X is said to be of finite type provided that a subset B...

A collection  of subsets of a set X is said to be of finite type provided that a subset B of X belongs to  if and only if every finite subset of B belongs to . Show that the Kuratowski lemma implies the following:

Lemma (Tukey, 1940). Let  be a collection of sets. If  is of finite type. then  has an element that is properly contained in no other element of .