axiomatic sets 10 (a formula)

In naive set theory, we wrote the set of real numbers in the interval $[0,1]$
like  $\left\{x\in\mathcal{R} | x\in [0,1]   \right\}$ .

The formula $x\in [0,1]$  is the property or the condition which the elements $x$ have.

A set is a collection of some objects.
(there are also some cases in which a collection is not a set and becomes a class. )

We have to state various kinds of properties which elements of a set have for defining the set.
It is a formula $P(x)$ .

IF $P(x)$ is a formula (a statement of properties of $x$ ),
then logical notations below can be only accepted in $P(x)$ ;

belongs to : $\in$
or : $\vee$
and : $\wedge$
not : $\neg$
If then : $\rightarrow$
for all : $\forall$
for any : $\exists$

Occasionally,  $E!(A)$ means " there exists only one A" .

Of course, new notations which have been derived by above are possible.
For example, $\subset$ , $\cup$ ,and so on.

We can also write $\neg(x\in a)$  $x\notin a$ .

This is not an axiom. It is a promise.