axiomatic sets 4 (empty set)

First axiom of ZFC is the existence of empty set.

It is stated by following logical formula;
\[ \exists x\forall y[y\notin x]  \]

Using A familiar expression,
\[ \phi=\left\{ \right\}=\left\{x ; \forall y\notin x\right\},  \]
or,
\[  \forall y\notin \phi \]

This axiom means three things. One is a set(e.g. empty set) always exists in any spaces,
two is there is a set which has no elements,
and three is any sets has empty set as a element

Namely, it is always true that if $x\in\phi$ , then any $y$ has the member $x$ .

As the number of elements of empty set is zero, $0$ can be used instead of $\phi$ .
Please note that $0$  is a set in the theory.

This axiom is proved by another axioms.
However, it is one of ZFC-axioms.