Set Builder Notation with quantifier

Question

The standard set builder notation is presented in many books as:

$\{x:P(x)$}

But this seems to make more sense:

$\{\forall x:P(x)$}

I've never seen it with a $\forall$ quantifier in front of the object $x$ though. I just want to make sure this is correct.

Answer 1

$A = \{x | P(x)\}$ simply declares that 'the set A is the set of all x, such that x satisfies the predicate P'. It is a shorthand way of writing the equivalent statement in logic, being $\forall x(x \in A \leftrightarrow P(x))$