If $C\subseteq B \subseteq A$ then $A\backslash C = (A\backslash B)\cup (B\backslash C)$ Is it true that if $C\subseteq B \subseteq A$ then $A\backslash C = (A\backslash B)\cup (B\backslash C)$? By drawing Venn diagrams this clearly seems so, but I can't seem to prove it.
Similarly, is it true that if $C\subseteq A$ and $C\subseteq B$ then $A\backslash C \subseteq(A\Delta B)\cup(B\backslash C)$?
If they are, what is the proof? Otherwise, what is a counterexample?

Let $x\in A-C$. Then $x\in A$ and $x\not\in C$. If $x\in B$, then $x\in B$ and $x\not\in C$, so $x\in B-C$, while if $x\not\in B$, then $x\in A$ and $x\not\in B$, so $x\in A-B$. This proves that $A-C\subset (A-B)\cup (B-C)$.
Now suppose that $C\subset B\subset A$. Let $y\in (A-B)\cup (B-C)$. If $y\in A-B$, then $y\in A$ and $y\not\in B$, so $y\in A$ and $y\not\in C$, so $y\in A-C$, while if $y\in B-C$, then $y\in B$ and $y\not\in C$, so $y\in A$ and $y\not\in C$, so $y\in A-C$. This proves that $(A-B)\cup (B-C)\subset A-C$.
