In "Union-free Hypergraphs and Probability Theory", the first theorem states that the upper bound on $f(n)$, which is defined to be
the maximum number of distinct subsets of an $n$-element set such that all the $f(n) \choose 2$ pairwise unions are distinct
is
$f(n) \leq 1 + 2^{(n+1)/2}$
Then they state that it's "an immediate consequence of ${f(n) \choose 2} \leq 2^n$." Can someone explain this "immediate consequence?
As far as I understand hypergraphs, the only upper bound I could come up with for $f(n)$ itself is $2^n$, so ${f(n) \choose 2} \leq 2^{(n-1)}(2^{n} - 1)$.