Axiom to prove $ \sqrt2 $ is irrational by contradiction.

Strictly speaking, we do not prove that $\sqrt 2$ even exists, i.e., we only prove

There is no rational number $x$ with $x^2=2$.

Similarly, we can prove

There is no rational number $x$ with $0\cdot x=1$.

But after such a proof, we would not say that "$\frac10$ is irrational". Instead, we say that $\frac 10$ is not defined. What is the difference?

Historically, $\sqrt 2$ appeard as length of the diagonal of a unit square, which by Pythagoras had the property that $x^2=2$. So the existence in some sense was not under doubt. But once we are given existence of a number with this property (or with another property $P$), there is no difference between "There is no rational number with property $P$" and "The (or any) number with property $P$ is irrational". Note that all non-real complex numbers, for example are irrational, hence questions about real or imaginary or complex can be ignored. If we know, say, from the fundamental theorem of algebra that $X^2-2$ has some complex root, then there is no need to first show that it is real before showing that it is not rational.