They didnt. Even in this panel he never once even referred to himself as the greatest swordsman.

Mihawk has never said verbatim "I am the greatest swordsman".

Every mention of him being such is literally stated by the world government/ oda/ and the onepiece verse itself. But mihawk himself has never said those words
Why does it matter if he said those exact words? No matter what, he still said he's the greatest.
 
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