Notation for non-empty subset To denote non-empty subsets, I repeatedly find myself writing $A\subset S, A\neq \emptyset$. Is there any established shorthand for this, you know, like $A\subset S$ can be seen as a shorthand for $A\subseteq S, A\neq S$?
 A: Try $\emptyset \neq A \subset S$. 
A: If I recall correctly, $A \in \mathcal{P}_{\geq 1}(S)$ is somewhat standard notation for "$A$ is a non-empty subset of $S$."
More generally, I think that $\mathcal{P}_{\geq \kappa}(S)$ is sometimes used to denote the collection of all subsets of $S$ with cardinality greater than or equal to $\kappa.$
A: Define 
$A \subset_\emptyset B \Leftrightarrow (A \subset B \wedge A \neq \emptyset)$. Then use $\subset_\emptyset$ instead of $\subset$.
A: Probably not, notations differ from author to author. Not even $A\subset S$ is unambiguous - it can mean proper subset or just subset. This means that if you're going to use a shorthand notation you would probably need to define the meaning of it. The mentioned $\mathcal P_{\ge 1}$ for example is nothing I've encountered, and the same applies to $\subseteq_{\emptyset}$.
Like it or not the notation $\emptyset\ne A\subset S$ or even $A\subset S, A\ne0$  is the closest thing to something that would be immediately understood and it's almost as compact as $A\subset_{\emptyset}S$, or $A\in\mathcal P_{\ge1}(S)$.
Normally you would have it in text and then you could write it out "non-empty $A\subset S$". As pointed out you should consider the context in the case as well. If you want to use $\emptyset\ne A\subset S$ you should use it in a context that calls for a statement, for example "assume that $\emptyset\ne A\subset S$".
