Set Difference with Empty Set is Self

Theorem
$$S - \varnothing = S$$

First, we have $$S - \varnothing \subseteq S$$ 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 $$S - \varnothing \subseteq S$$ and $$S \subseteq S - \varnothing$$.

Thus by the definition of set equality, $$S - \varnothing = S$$.