Definition:Existential Quantifier

The symbol $$\exists$$ is called the existential quantifier. It expresses the fact that, in a particular universe of discourse, there exists (at least one) object having a particular property.

That is:

$$\exists x$$ means "There exists at least one object $$x$$ such that ...".

Some authors call this the particular quantifier.

An existential statement is one which expresses the existence of at least one object which has a particular property.

There are variants of this symbol:

$$\exists !$$ means "there exists uniquely", or, "there is one and only one".

$$\exists_n$$ means "there exist exactly $$n$$".

Thus $$\exists_1$$ means the same thing as $$\exists !$$.