Prove $A setminus (A setminus (A setminus B)) = A setminus B.$

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The question is as is in the title. I am able to show that $A setminus (A setminus (A setminus B)) subseteq A setminus B$ but I am stuck on showing that $A setminus B subseteq A setminus (A setminus (A setminus B))$.

Am I doing something wrong? Any suggestions will help.

Here is what I have so far:

$textbf{Proof:}$

To show that $A backslash (A backslash (A backslash B)) = A backslash B$, we need to show that $A backslash (A backslash (A backslash B)) subseteq A backslash B$ and $A backslash B subseteq A backslash (A backslash (A backslash B))$.

We first show that $A backslash (A backslash (A backslash B)) subseteq A backslash B$.

Let $a in A backslash (A backslash (A backslash B))$.

$implies a in A cap (A backslash (A backslash B))^c$

$implies a in A cap (A cap (A backslash B)^c)^c$

$implies a in A cap (A cap (A cap B^c)^c)^c$

$implies a in A cap (A^c cup (A cap B^c))$

$implies a in A cap ((A^c cup A) cap (A^c cup B^c))$

$implies a in A cap (U cap (A^c cup B^c))$

$implies a in A cap (A^c cup B^c)$

$implies a in (A cap A^c) cup (A cap B^c)$

$implies a in emptyset cup (A cap B^c)$

$implies a in (A cap B^c)$

$implies a in (A backslash B)$

Next, we show that $A backslash B subseteq A backslash (A backslash (A backslash B))$.

Let $a in A backslash B$.

$implies a in A backslash (A backslash B)^c$

$implies a in A backslash (A backslash (A backslash B)^c)^c$

$implies a in A backslash (A cap (A backslash B))^c$

$implies a in A backslash (A^c cup (A backslash B)^c)$

$implies a in A backslash (A^c cup (A^c cup B))$