I've been having a hard time understanding why $\emptyset\notin\{\emptyset,\{\emptyset\}\}$. Why isn't it an element in that set? Also why is $\{\emptyset\}\in\{\emptyset\}$ true? I know that $\{0\}\notin\{0\}$ so why is that one true and this false?

Determine whether these statements are true or false.

1. $\emptyset\in\{\emptyset\}$
2. $\emptyset\in\{\emptyset,\{\emptyset\}\}$
3. $\{\emptyset\}\in\{\emptyset\}$
4. $\{\emptyset\}\in\{\{\emptyset\}\}$
5. $\{\emptyset\}\subset\{\emptyset,\{\emptyset\}\}$
6. $\{\{\emptyset\}\}\subset\{\emptyset,\{\emptyset\}\}$
7. $\{\{\emptyset\}\}\subset\{\{\emptyset\},\{\emptyset\}\}$