Proper subset - Knowlpedia

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).

Remarks

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 $A\neq \varnothing$ .