I somehow got a weird result when proving it.
Suppose $A \cup B = U$. Then by de Morgan's Law,
$A^c \cap B^c = \emptyset$.
Taking the intersection of both sides with $A$,
$A\cap A^c \cap B^c = A \cap \emptyset = \emptyset$.
$U \cap B^c = \emptyset$
$B^c = \emptyset$?
$U$ is the universal set
Where did I go wrong here?