# Is it false or it cannot be proven to be true

On a re-reading of D.J.Vellemann's book - "How to prove it" (2nd edition, pg. 69), it reads

It should be clear that if $A =\varnothing$, then $\exists x \in A P(x)$ will be false no matter what the statement $P(x)$ is. There can be nothing in $A$ that, when plugged in for $x$, makes $P(x)$ come out true as well.

This second line actually means that suppose $P(x)$ is true, then we can't find anything that proves it to be true. In other words, one cannot prove that $P(x)$ is true rather than proving $P(x)$ is false. Am I right in this reasoning?

It is not a question about whether $$P(x)$$ is true or false, it is whether $$\exists\ x\in\varnothing\quad P(x)$$ is true or false, and this is a different statement. This statement says that there is an element $x$ in $\varnothing$ such that...
...well, I don't need to go any further - there is no element at all in $\varnothing$, so whatever I was about to say, it will be a false statement because it asserts, among other things, that there is an element in $\varnothing$.
$\exists x \in S : P(x)$ is a shorthand for $\exists x : x \in S \land P(x)$. So (as David explains well) when $S = \emptyset$ it is always false.