How can I prove (or state that this statement is wrong) "For any real number $x$, if $x < x$ then $x$ is a rational number."

Question

I feel really confused and I don't know how should I prove it. By common sense, it seems like there are no real numbers satisfying $x < x$.

Answer 1

Hint: If $A$ is not true, then $A\implies B$ is always true, not matter what $B$ is.

Answer 2

$\{x:x \in \mathbb{R}$, and $x <x\}=\emptyset$.

There is no element $x \in \emptyset$.

The statement is vacuously true.