Union with Empty Set

Theorem
$$S \cup \varnothing = S$$

Proof
$$S \subseteq S$$ Subset of Itself

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

$$\Longrightarrow S \cup \varnothing \subseteq S$$ Smallest Union

$$S \subseteq S \cup \varnothing$$ Subset of Union

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