Set-builder Notation

Question

In set-builder notation we describe a set in the following way:

$A=\left\{x:\phi (x)\right\}$

Is it correct to say the following?

1. Fix any $x_{0}\in X$
2. Evaluate the predicate $\phi(x_{0})$
3. $x_{0}\in A$ if and only if $\phi(x_{0})$ is true

I am not sure about the "if and only if" part.

Thank you!

Answer 1

We can read A as " The set of all $x$ such that $\phi (x) $ is true". If $x \in A$ then we know that $\phi(x)$ is true. If $\phi(y)$ is true, then $y$ must belong to the set of all $x$ such that $\phi(x)$ is true. And so we have that $x \in A$ iff $\phi(x)$ is true.