Intersection with Empty Set

Theorem
$$S \cap \varnothing = \varnothing$$

Proof
$$S \cap \varnothing \subseteq \varnothing$$ Intersection Subset

$$\varnothing \subseteq S \cap \varnothing$$ Empty Set Subset of All

$$\Longrightarrow S \cap \varnothing = \varnothing$$ Definition of Set Equality