Incorrect reasons for rejection, what next? Recently I received a rejection to my submission of an article to a journal. Despite three positive reviews, an additional reviewer wrote a short response with an example to claim that a proof in my paper was wrong. The editor claimed to "carefully" check the example and for this reason rejected my paper (no resubmission).

Now, this example is not a counterexample, which I can show with a single sentence (it doesn't satisfy the property I assumed in my theorem). So what should I do?
Ok—Your paper was rejected because a reviewer made a mistake. I'll use my powers of Bayesian Induction to guess the editor just defaults to this reviewer, and there's not academic fraud issue. Everyone in academia has a drawer in their desk of decent papers that didn't get published just because. 

You already emailed your advisor. You took the first step. That's good. I hope you didn't imply the editor's against you, but if you did, so be it. 

Here's what you should you do now:


Go into the bathroom, take a deep breath, look in the mirror, and say: "this is going to be OK." Good papers get initially rejected for stupid reasons all the time.
Show the statement of the theorem, the purported counterexample, and your one-sentence refutation to your advisor or another colleague and get their reaction, to check whether you might have misunderstood the issue.
Did the editor communicate directly with you? How did they communicate with you? If it was by email, write an email back. Write a three-sentence email about why the reviewer's counterexample isn't a counterexample. Show your proposed email to your advisor for feedback before sending it to the editor.
You will wait to get a yes-no answer. 
If you still get rejected, submit the paper to another journal.
If all else fails, put the paper up on arxiv and try to present it at a conference.