Can we deduce two contradictory statements from a wrong statement? In the proof that $\sqrt{p}$ is irrational where $p$ is a prime number:
We first assume $\sqrt{p}$ is rational.
From this we deduce $\sqrt{p}=\dfrac{a}{b}$, where $a$ and $b$ are co-prime.
Then using other reasonings we deduce $a$ and $b$ are not co-prime.
That is from the wrong statement (i.e. $\sqrt{p}$ is rational), we deduced two contradictory statements.
From a statement, if we apply correct reasonings, we may deduce two results which contradict each other. How can this be reasonable? Can anyone explain with simple examples?
Anyway; from a statement, if we apply correct reasonings, we may deduce a result which contradicts another established result. This seems reasonable to me. And this shows that the statement is wrong.

The statement that $\sqrt{p}$ (with $p$ a prime number) is rational is not just a false statement, but a statement that contradicts the basic axioms for the real numbers. Indeed, it is from the statement that $\sqrt{p}$ is rational together with those axioms that you derive an explicit contradiction.
