Suppose $A$ and $B$ are sets. Prove $A\subseteq B$ iff $A-B=\emptyset$.
Assume $A\subseteq B$. This means that if $a\in A$ then $a\in B$. Since $A-B$ means all $a\in A$ such that $a\not\in B$, we can see that $A-B=\emptyset$.
Now assume $A-B=\emptyset$. This means that there are no $a\in A$ such that $a\not\in B$. This implies that if $a\in A$ then $a\in B$. Therefore $A\subseteq B$.
Since we have shown that $(A\subseteq B)\Rightarrow (A-B=\emptyset )\wedge (A-B=\emptyset )\Rightarrow (A\subseteq B)$, we can conclude that $A\subseteq B$ iff $A-B=\emptyset$.
$\blacksquare$
Though I can't pinpoint anywhere I went wrong here, I can't help but feel there's a hole in my reasoning somewhere... is this a valid proof?