What's wrong with the following proof of the theorem? I am reading How to prove it.
In Section 3.2, it asked us what's wrong with the following proof of the theorem?
Suppose $x$ and $y$ are real numbers and $x+y = 10$. Then $x \neq 3$ and $y \neq 8$.
Proof:
Suppose the conclusion of the theorem is false. Then $x=3$ and $y = 8$. But then $x+y = 11$, which contradicts the given information that $x+y = 10$. Therefore the conclusion must be true.
I can feel that the mistake is in $x=3$ and $y=8$. Should it be $x=3$ or $y=8$  since it negates $x=3$ and $y=8$?
But I intuitively think the theorem is correct because $x$ cannot be 3 and $y$ cannot be 8 at the same time if $x + y = 10$ is true. Can anyone explain?

It should be $x=3$ or $x=8,$ which does not give a contradiction.
The theorem is wrong.
Try $x=3$ and $y=7$.
Also, try $y=8$ and $x=2$.
