Differences in $\varnothing$, {$\varnothing$}, and $\subseteq$ What is the difference between
{$\varnothing$} $\subseteq$ {$\varnothing$, {$\varnothing$}},
{{$\varnothing$}} $\subseteq$ {$\varnothing$, {$\varnothing$}},
and
{{$\varnothing$}} $\subseteq$ {{$\varnothing$}, {$\varnothing$}},
where $\varnothing$ is the empty set?
 A: $\varnothing$ is the empty set; it has no elements.
$\{\varnothing\}$ is a set that has exactly one element; that element is $\varnothing$ (a bag that contains an empty bag is not itself empty).
$\bigl\{\{\varnothing\}\bigr\}$ is a set whose only element is the set whose only element is $\varnothing$. It is different from $\varnothing$ (which has no elements), and from $\{\varnothing\}$ (which has a single element which has no elements, whereas the single element of $\bigl\{\{\varnothing\}\bigr\}$ does have elements). 
And $\bigl\{ \{\varnothing\},\{\varnothing\}\bigr\} = \bigl\{ \{\varnothing\}\bigr\}$, by the Axiom of Extensionality: two sets $A$ and $B$ are equal if and only if for every $x$, $(x\in A\leftrightarrow x\in B)$ which holds here.
So the first statement says that $\{\varnothing\}$ is a subset of $\bigl\{\varnothing,\{\varnothing\}\bigr\}$; essentially, that $\varnothing$ is an element of the set on the right; true.
The second statement says that $\bigl\{\{\varnothing\}\bigr\}$ is a subset of $\bigl\{\varnothing,\{\varnothing\}\bigr\}$; again, essentially that $\{\varnothing\}$ is an element of the set on the right; also true, but a different statement from the first (it refers to different elements).
The third statement says that $\bigl\{\{\varnothing\}\bigr\}$ is a subset of $\bigl\{\{\varnothing\},\{\varnothing\}\bigr\} = \bigl\{ \{\varnothing\}\bigr\}$; that is, that $\{\varnothing\}$ is an element of $\bigl\{\{\varnothing\}\bigr\}$. Also true.
