Brazilian Jiu-Jitsu

TLDR: How do I gain the respect of a student who thinks they know better than me?

I'm a 22-year-old purple belt who has been training for nearly 5 years at a 10th Planet gym, I include these details because they are relevant don't worry! I've recently been teaching a few classes when my coach feels sick (or lazy).

Whilst teaching a few days ago, I had a tricky situation. We have a student who is a roughly 32 y/o blue belt MMA fighter. He's a typical MMA fighter in his style and has been training for 6/7 years. He mostly does MMA classes and not BJJ ones specifically, he also doesn't really use 10th Planet techniques, he mostly just pins people. He always asks our head coach about being promoted and speaks disparagingly of people who have been promoted ahead of him, myself included.

Whilst I was teaching a technique, someone asked a question, and he interrupted me to answer. Most annoyingly, what he said was wrong, and not what we were teaching. I tried to be diplomatic and explain that what he said could be a possible technique from the position. but it is not high percentage, and more importantly, isn't the technique that I was demonstrating. He remained insistent that what he said was correct and that it was better than what I was teaching. So I said that he can show me it whilst people were drilling or whilst we were rolling later because it didn't seem right to outright dismiss him.

I then approached him whilst people were practising the technique, and he didn't want to go through it with me. I feel as though he just wanted to correct me whilst I was teaching, or just that he wanted to get his two cents in. I get the impression that he doesn't respect me because he thinks I was unfairly promoted ahead of him.

What can I do in future to mitigate this sort of situation or prevent it?