How can it be that the empty set is a subset of every set but not an element of every set?
I've understood that the empty set must be a subset of every set because if it were not a subset of every set then the statement $\exists x : x \in \emptyset \land x \notin M$, would need to be ture, but since the empty set has no elements this statement is a contradiction.
But how can it be that the empty set is not element of every set, if it is a subset of every set?
I also understood that it cannot be an element of itself, otherwise it wouldn't be the empty set anymore, but why is it not an element of every non-empty set? If the empty set is subset of every set, than it also would need to be a subset of itself ($\emptyset \subset \emptyset$), how can that be?