For any set $A$, both $\emptyset \subseteq A$ and $\emptyset \in A$ hold.

False

False. $\emptyset \subseteq A$ always holds, vacuously, since there is no element of $\emptyset$ to find missing from $A$. But $\emptyset \in A$ requires the empty set to be listed as a member, which is true only for sets that happen to contain it, such as $\{\emptyset\}$. Inclusion and membership are different relations.