Intersection with Complement is Empty iff Subset

Theorem

 * $S \subseteq T \iff S \cap \map \complement T = \O$

where:
 * $S \subseteq T$ denotes that $S$ is a subset of $T$
 * $S \cap T$ denotes the intersection of $S$ and $T$
 * $\O$ denotes the empty set
 * $\complement$ denotes set complement.

Also presented as
Some sources present this as an alternative definition of a subset:
 * Set $A$ is a subset of set $B$ if set $A$ contains no member that is not also in set $B$

but this is not mainstream.

Also see

 * Complement Union with Superset is Universe