Is there a name for a function that tests a set is a subset of another, but is not empty?

Question

I would say $A$ is a subset of $B$ if $A$ contained $(X,Y,Z)$ and $B$ contained $(W,X,Y,Z)$.

But if $A$ contains the null set, and $B$ something, (or nothing?) and I still need the result to be false, is there a name for this?

if $F = A \subset B$,

\begin{array} {|r|r|r|} \hline A &B &F(A,B)\\ \hline & &False\\ \hline &4 &False\\ \hline 3 &&False \\ \hline 3 &3&True \\ \hline 3&3,4&True \\ \hline 3,4&3&False \\ \hline \end{array}

Answer 1

Non-empty subset? $$\quad\emptyset\subsetneq A\subseteq B$$

Or, equivalently (and mayhap easier to read): $~0\neq A\subseteq B$