# how to proof ${ NPC \bigcap CO-NPC \ne \varnothing then NP = P ? }$

how proof $${\ \ NPC \ \ \bigcap \ \ CO-NPC \ne \varnothing }$$
then $${NP = P ? }$$

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I don't think you're likely to find any such proof. Given our current level of knowledge, as far as we know it is possible that $$\textsf{P} \ne \textsf{NP}$$ but $$\textsf{NP} = \textsf{co-NP}$$ (we cannot prove otherwise). If that were true, then we'd have $$\textsf{NPC} = \textsf{co-NPC}$$ (and thus $$\textsf{NPC} \cap \textsf{co-NPC} \ne \emptyset$$) yet $$\textsf{P} \ne \textsf{NP}$$.