Is the P vs NP problem, an NP or co-NP problem?

Question

A few years ago, a youtube channel named hackerdashery, made an extraordinary youtube video explaining P vs NP, in a semi-vulgarized way : https://www.youtube.com/watch?v=YX40hbAHx3s

At 7 minutes and 56 seconds however, he talks about why it is so hard to prove that P $=$ NP or that P $\neq$ NP, and states that proving things is in fact an NP problem (in the video however, he writes down that it is actually a co-NP problem).

Thus, is proving things, in particular P $=$ NP or P $\neq$ NP, an NP or co-NP problem? If so, why? And where could I find more on this?