Two sets A and B are the same, if
$ \displaystyle \forall X: [X \in A \Leftrightarrow X \in B] \land [A \in X \Leftrightarrow B \in X]$
Ehm. I know one explanation that says, "two sets are equal, if they own the same elements". If so, why do we need the second part of the definition? If I read this, I see: "Two sets are equal, if they own the same elements and are elements of some other thing". But then, they are elements of that that is an element of themselves. What are these circles of mind?
Can somebody explain me the sacral meaning of this second part? Thanks.