Let $A$ be an arbitrary non-empty set. Then, by definition of the empty set, $\emptyset\subseteq A$. Consider now a non-empty subset $B\subseteq A$. In order to formally specify that the subset $B$ is non-empty, should I write $B\subseteq A\backslash \emptyset$ or $B\subseteq A\backslash\{\emptyset\}$?
To me, writing $B\subseteq A\backslash \emptyset$ seems more natural, for the emptyset is itself a set and thus the curly brackets seem redundant and incorrect. However, $B\subseteq A\backslash\{\emptyset\}$ also seems to have some intuitive appeal. Can you please clarify which is the correct way of writing it and why?
Neither is correct: let $A:=\{1,2,3\}$; you can write $A-\{1,2\}$ but not "$A-\{\emptyset\}$" since $\emptyset \notin A$.
A good notation is $$(B\subset A \text{ and } B\neq \emptyset )\iff B\in \mathcal{P}(A)- \{\emptyset\}$$where $\mathcal{P}(A)$ is the power set of $A$.