I have a following problem:

Let $P^{SAT[1]}$ be a class of problems decidable by a deterministic polynomial Turing Machines with SAT oracle. (only one question to oracle).

Assume that: $coNP \neq NP \neq P$

Decide if $coNP \cup NP \neq P^{SAT[1]}$ or $coNP \cup NP = P^{SAT[1]}$.

I know how to show that $coNP \cup NP \subseteq P^{SAT[1]}$.

But I have no idea what about inverse direction. Any tips?


Hint (noting the homework tag on the other post): $P^{SAT}$, by the Cook-Levin theorem, is the same as an $NP$ oracle. Given that $\text{co-}NP \neq NP$, what does this imply?


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