From Bishop's constructive analysis book I know for some real numbers $x$ and $y$ at most one of the statements $x=y$, $x \ne y$ is true.
Can I also show constructively that exactly one of the statements is true (i.e. a need to show that at least one of the statements is true)? My intuition says this is not possible since this would be some kind of a $A\vee\lnot A$ statement.