It is an open question if NP $\neq$ Co-NP but if the conjecture were proved, this would mean that P $\neq$ NP because P is closed under complement. Now a fact that fails to enter my head is the following:
We take 3SAT which we know is NP-complete and then we take 3TAUTOLOGY which is Co-NP-complete. The last formal language is a subset of the first, however it seems to have certain structural properties that make it (more) difficult to recognize. it is known that the certificate for SAT consists in a binary distribution of the literals that satisfies the formula but the certificate for a TAUTOLOGY must necessarily contain ALL the distributions of the variables, which must ALL satisfy the formula, otherwise we could not ascertain in any way that it is a tautology.
Therefore, based on this fact, verifying belonging to the 3SAT set by having a certificate requires time polynomial to the input but verifying membership in the set 3TAUTOLOGY no. Why is this fact not enough to show that NP $\neq$ Co-NP?
Is the reasoning correct up to this point? if so, should it still be formally demonstrated? Is it in doing this that lies the immense difficulty?