Set Difference with Non-Empty Proper Subset is Non-Empty Proper Subset

Theorem
Let $S$ be a set.

Let $T \subsetneq S$ be a proper subset of $S$. Let $S \setminus T$ denote the set difference between $S$ and $T$.

Then:


 * $S \setminus T$ is a proper subset of $S$

Proof
From Set Difference is Subset:
 * $S \setminus T \subseteq S$

From Set Difference with Proper Subset:
 * $S \setminus T \ne \O$

By definition of a proper subset:
 * $\exists x \in T$

From Set Difference and Intersection are Disjoint:
 * $\paren{S \setminus T} \cap \paren{S \cap T} = \O$

From Intersection with Subset is Subset:
 * $S \cap T = T$

By definition of set intersection:
 * $x \notin S \setminus T$

By definition of subset:
 * $x \in S$

By definition of set equality:
 * $S \setminus T \ne S$

It follows that $S \setminus T$ is a proper subset by definition.