Prove that the cardinality of the intersection of the power sets of two sets is of the form $2^n$ for some positive $n$.
My Thoughts:
Let the 2 sets be A and B
$$ A = \{1,2,3,4,5\} $$
$$ B = \{3,4,5,6,7\} $$
so,
$A\cap B = \{3, 4, 5\}$
$P(A\cap B ) = \{\{\varnothing\} ,\{3\},\{4\},\{5\},\{3, 4\},\{3 ,5\},\{4, 5\}, \{3, 4, 5\}\}$
$|P (A \cap B )| = 2^n$, where $n$ is the number of elements in the set
$|P (A\cap B )| = 2^3 = 8$
I proved 1 case. But how do I prove for all cases?