Empty Set is Subset of All Sets/Proof 1

Theorem
The empty set $\varnothing$ is a subset of every set (including itself).

That is:


 * $\forall S: \varnothing \subseteq S$

Proof
By the definition of subset, $\varnothing \subseteq S$ means:


 * $\forall x: \left({x \in \varnothing \implies x \in S}\right)$

By the definition of the empty set:


 * $\forall x: \neg \left({ x \in \varnothing }\right)$

Thus $\varnothing \subseteq S$ is vacuously true.