Subsets of sets containing empty set Why is $\{\emptyset\}$ not a subset of $\{\{\emptyset\}\}$?
It contains this element, but why is it not a subset?
 A: It is not a subset because its element $\emptyset$ does not belong to the set $\{\{\emptyset\}\}$. 
A: The element $\{\emptyset\}$ and the element $\emptyset$ are different.
$\{\emptyset\}$ is an element of $\{\{\emptyset\}\}$, whereas $\emptyset$ is not.
A subset is a set whose every element is also a part of the given set.
Thus, the subsets of $\{\{\emptyset\}\}$ are $\{\{\emptyset\}\}$ and the empty set $\{\}$, also denoted by $\emptyset$.
EDIT:
In one sentence, (Thanks to @Henry)
$\{\{∅\}\}$  has a single element $\{∅\}$ and two subsets $\{\{∅\}\}$ and $∅$, while $\{∅\}$ has a single element $∅$ and two subsets $\{∅\}$ and $∅$.
A: You can think of sets like plastic bags if you want; the empty set is just a plastic bag with nothing in it, $\{\emptyset\}$ is a plastic bag with another plastic bag in it, and $\{\{\emptyset\}\}$ is three layers of plastic bags.
The element relation $A\in B$ means that you could open up bag $B$ and take out $A$.
The subset relation $A\subset B$ means that every object that you could directly take out of $A$ can also be directly taken out of $B$.
So, look at $\{\emptyset\}$. You can "open it up and" take out $\emptyset$, but you can't do that with $\{\{\emptyset\}\}$. Therefore, $\{\emptyset\}\not\subset \{\{\emptyset\}\}$.
