# $∅\in\{∅,\{∅\}\}$ is false but $∅\in\{∅\}$ is true… why?

I could be wrong, but I learned that $∅\in\{∅,\{∅\}\}$ is false but $∅\in\{∅\}$ is true... why is that? I would think that both of these would have to be true. If $∅\in\{∅\}$ is true wouldn't $\{∅\}$ be an element in $\{∅,\{∅\}\}$?

• $a\in\{a,\{a\}\}$ for all $a$, including $a=\emptyset$. – Lord Shark the Unknown Sep 1 '18 at 19:10
• Both are true. Maybe you meant $\emptyset \in \{\{ \emptyset \} \}$ is false – leibnewtz Sep 1 '18 at 19:11
• In ordinal notation the first statement is $0\in 2$, while the second is $0\in 1$. Both are true, but it's theoretically possible you got the first statement mixed up with one that actually is false, such as $0^+=2$. – J.G. Sep 1 '18 at 19:14
• Several similar questions have been posted here; a very detailed answer by Daniel is here. – Dietrich Burde Sep 1 '18 at 19:20
• Thank you all for that clarification! I thought I was going mad. – maximusg Sep 1 '18 at 19:26