Set Difference with Superset is Empty Set

From ProofWiki
Jump to navigation Jump to search


$S \subseteq T \iff S \setminus T = \O$


$S \subseteq T$ denotes that $S$ is a subset of $T$
$S \setminus T$ denotes the set difference between $S$ and $T$
$\O$ denotes the empty set.


\(\ds \) \(\) \(\ds S \setminus T = \O\)
\(\ds \) \(\leadstoandfrom\) \(\ds \neg \paren {\exists x: x \in S \land x \notin T}\) Definition of Empty Set
\(\ds \) \(\leadstoandfrom\) \(\ds \forall x: \neg \paren {x \in S \land x \notin T}\) De Morgan's Laws (Predicate Logic)
\(\ds \) \(\leadstoandfrom\) \(\ds \forall x: x \notin S \lor x \in T\) De Morgan's Laws: Disjunction of Negations
\(\ds \) \(\leadstoandfrom\) \(\ds \forall x: x \in S \implies x \in T\) Rule of Material Implication
\(\ds \) \(\leadstoandfrom\) \(\ds S \subseteq T\) Definition of Subset