Definition:Proper Subset/Also defined as

Proper Subset: Also defined as

Some authors require that $S \ne \O$ for $S$ to be a proper subset of $T$.

Hence, under this convention, $S$ is a proper subset of $T$ if and only if

$\O \subsetneqq S \subsetneqq T$

It is wise to be aware of which definition is in use.

Sources

• 1966: Richard A. Dean: Elements of Abstract Algebra ... (previous) ... (next): $\S 0.2$. Sets
• 1971: Allan Clark: Elements of Abstract Algebra ... (previous) ... (next): Chapter $1$: The Notation and Terminology of Set Theory: $\S 3$
• 1975: Bert Mendelson: Introduction to Topology (3rd ed.) ... (previous) ... (next): Chapter $1$: Theory of Sets: $\S 2$: Sets and Subsets