I'm also interested in the converse of this: does $A \in B \implies B \not\subset A $?

I can give examples where the first one holds: (e.g. $ A = \{1,2,3\}, B= \{1,2,3,4\}$), or the second one holds ($A=\{1\}, B=\{\{1\}\} $), but I can't come up with counterexamples for either of them.