Set Theory and Equality Let $A$ and $X$ be sets. Show that $X\setminus(X\setminus A)\subseteq A$, and that equality holds if and only if $A\subseteq X$. 
I understand why this holds but am not sure how to 'show' this. Any advice would be appreciated. 

Define that $A\subseteq X  \,\,\Leftrightarrow\,\, \forall x\in  A \Rightarrow x\in X$
Pain attetion that: $X\backslash A\Leftrightarrow x\in X$ and $x\notin A$.
and, by De morgan, $(X\backslash A)^c \Leftrightarrow x\notin X$ or $x\in A.$
$X\backslash(X\backslash A)\Leftrightarrow x \in X \,\,\text{and}\,\, x\notin (X\backslash A)\Rightarrow x\in X$ and $x\in (X\backslash A)^c\Rightarrow x\in X$ and $(x\notin X$ or $x\in A)\Rightarrow x\in X$ and $x\in A \Rightarrow x\in A$.
For the equation I need to prove the inverse (remember that $A\subseteq X$ is true):
$A\subseteq X\backslash(X\backslash A)$.
