Maximal intersecting families choose one member of every complementary pair
= Maximal intersecting families choose one member of every complementary pair
A maximal intersecting family is an <up-set>. It cannot contain both a set and its complement. If neither were present, maximality would supply a member disjoint from the first set, and upward closure would then include its complement. Thus precisely one member of each complementary pair occurs, so the family has $2^{n-1}$ members for $n\geq1$.