Proper subset

A proper subset of a set $B$ is a set $A$ such that $A\subseteq B$ and $A\neq B$. One writes $A\subsetneq B$ (or $A\subset B$ in contexts where ambiguity is avoided).

Proper inclusion strengthens the subset relation by excluding equality, and it often appears when describing strict containments such as $A\subsetneq A\cup\{x\}$.

Examples:

• $\{1,2\}\subsetneq \{1,2,3\}$.
• For any set $A$, $\varnothing\subsetneq A$ holds exactly when $A\neq \varnothing$.