The power set of a set S, P(S) (or sometimes 2^S) is the set of all subsets of S including both the empty set and the set itself.

To bridge the gap with programming, make a map f: S -> bool which represents our predicate.

all(f, S) => either S is empty or for all elements s in S, f(s) = True.

Now make f work on sets as well as individual values. f({x, y}) means True if f(x) and f(y) are True, False otherwise.

all(f, S) => all(f, P(S))

If we take the opposite and define all(f, {}) = False then this doesn't work and in addition all(f, P(S)) = False for all sets S.