Is {∅} ⊆ ∅? Set theory

Question

Is {∅} ⊆ ∅? I'm guessing no, as {∅} is a subset of {∅} but not of ∅. But I'm having doubts as ∅ is the only element in {∅} and ∅ is ∅.

Answer 1

From the definition it follows that the only subset of the empty set is itself. By definition $\{ \phi \} \neq \phi$. So $\{ \phi \}$ is not a subset of $\phi$.

Answer 2

Indeed $\{ \emptyset \} \neq \emptyset$ since it has one element : $\emptyset$

Answer 3

It is indeed false. For a nifty application of this investigate von Neumann ordinals. The naturals are defined as $\emptyset,\{\emptyset\},\{\emptyset,\{\emptyset\}\},\dots$

Answer 4

The set $\{\varnothing\}$ has one element i.e. $\varnothing$ while the set $\varnothing=\{ \}$ has no element. So the set inclusion doesn't hold.