Set Difference with Empty Set is Self

Theorem
The set difference between a set and the empty set is the set itself:
 * $S \setminus \varnothing = S$

Proof
First, we have $S \setminus \varnothing \subseteq S$ from Set Difference Subset.

Next, we first note that $\forall x \in S: x \notin \varnothing$ from the definition of the empty set.

Let $x \in S$. Thus:

Thus we have $S \setminus \varnothing \subseteq S$ and $S \subseteq S \setminus \varnothing$.

So by the definition of set equality, $S \setminus \varnothing = S$.