Language in NPC and CoNP

Question

A few days ago I had a test that I failed to pass, and it had a question that I failed to do.

the question:

given:

$A \in NPC$

$A \in CoNP$

Determine which of the following statements is correct:

1. $P\neq NP$
2. $P\neq CoNP$
3. $NP\neq CoNP$
4. $NP=CoNP$
5. None of the above claims are true.

My idea to solve this, is to choose a language $B \in P$. From language $B$ it is possible to make a reduction to both problems to $CoNP$ and $NPC$. And take the complementary B language, $B^{'}$, which also belongs to the 2 groups.

Because B and B complement an identity then, it is possible to get that $NP = CoNP$ and $NPC = CoNPC$ , but I do not know if I am right in this solution.

I think 4 is the correct answer, but I do not know why the other answers are incorrect.

Answer 1

Let $L\in NP$. Thus, $L\le_p A$. Since $A\in coNP$, then $L\in coNP$. Hence, $NP\subseteq coNP$.

Now, let $L\in coNP$. Thus, $\overline{L} \in NP$ and therefore $\overline{L}\le_p A$. From reduction properties, we know that $L\le_p \overline{A}$ holds as well. Now, since $A\in coNP$ then $\overline{A}\in NP$. Hence, $L\in NP$, and therefore we get that $coNP\subseteq NP$

Now we can conclude that $NP\subseteq coNP$ and $coNP\subseteq NP$ and hence $NP=coNP$.