Sorry but I don't think I can know, since it's a definition. Please tell me. I don't think that $0=\emptyset\,$ since I distinguish between empty set and the value $0$. Do all sets, even the empty set, have infitinite emptiness e.g. do all sets including the empty set contain infinitely many empty sets?
Let $A$ and $B$ be sets. If every element $a\in A$ is also an element of $B$, then $A\subseteq B$.
Flip that around and you get
If $A\not\subseteq B$, then there exists some element $x\in A$ such that $x\notin B$.
If $A$ is the empty set, there are no $x$s in $A$, so in particular there are no $x$s in $A$ that are not in $B$. Thus $A\not\subseteq B$ can't be true. Furthermore, note that we haven't used any property of $B$ in the previous line, so this applies to every set $B$, including $B=\emptyset$.
(From a wider standpoint, you can think of the empty set as the set for which $x\in \emptyset\implies P$ is true for every statement $P$. For example, every $x$ in the empty set is orange; also, every $x$ in the emptyset is not orange. There is no contradiction in either of these statements because there are no $x$'s which could provide counterexamples.)
The empty set is subset of the empty set, as every element of the empty set is an element of the empty set. But $0$ is not in the empty set.
$A \subseteq B$ when $x\in A \implies x\in B$. As $x\in A \iff x\in A$ we see that $A \subseteq A$ is always true, when $A$ is a set.
A value is a value not a set, sometimes $0$ is defined as the empty set but then $0$ is the empty set and not the number.
This happens for example in category theory, as you are only interested in abstract sets, and all sets of the same cardinality are in a sense the same, you just title finite sets by their cardinality.
Because the reflexivity of $\subseteq$, for all $A$ set $A \subseteq A$ is true. For $A = \emptyset$ we have that $\emptyset \subseteq \emptyset$, so the empty set is a subset of itself.
The intersection of two sets is a subset of each of the original sets. So if $\phi$ is empty set and A is any set then $ \phi\cap A $ is $\phi$ which means $\phi$ is subset of A(which is any set) and $\phi$ is a subset of $\phi$ which implies empty set is subset of itself.
An empty subset can be selected from any set, including the empty set itself, using selection criteria that cannot be satisfied, e.g. selecting those elements $x$ such that $x\neq x$.
See my formal proof (in DC Proof format) at: http://dcproof.com/ExistenseOfNullSet.htm
There is only one empty set. If $\phi_1$ and $\phi_2$ are empty sets, then $\ \phi_1 = \phi_2$.
See my formal proof (in DC Proof format) at: http://dcproof.com/UniquenessOfNullSet.htm
Edit: For any set $X$ we can select an empty subset $S$ such that:
$\forall a:[a\in S\iff a\in X\land a\neq a]$
$\forall a:[a\in S\iff a\in X\land a\notin X]$