Assume that you are given "For every x∈S, property P(x) holds". You can say that the statement is false only if you can produce an element x such that the property is not satisfied. But, there exists no such x to claim that. Hence "ALL evaluates to TRUE if inner query returns no tuples".
Now similarly you are given "There exists x∈S such that property P(x) holds". Now this statement can only be asserted to be true when you can pick up an element x to show that this property is indeed satisfied. But since there is no such element "ANY evaluates to FALSE if inner query returns no tuples".