TITLE: Subsets of sets containing empty set
QUESTION [7 upvotes]: Why is $\{\emptyset\}$ not a subset of $\{\{\emptyset\}\}$?
It contains this element, but why is it not a subset?

REPLY [10 votes]: 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, $\{\emptyset\}\not\subset \{\{\emptyset\}\}$.