The intersection of $3$ set is empty, would the intersection of $4$ sets be empty? Let me clarify some more.
Let's say we have four sets $A,B,C,$ and $D$.
If the intersection of any three sets is empty, by default is the intersection of all four sets empty?

Yes. $A\cap B\cap C\cap D=A\cap B\cap(C\cap D)$ is the intersection of the three sets $A$ and $B$ and $C\cap D$, so it is empty.
