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The phrase "they do not exist" is grammatically correct and can be used in written English.
It is typically used to express that something or someone does not exist in reality. Example: "Despite rumors that unicorns roam the forest, the truth is that they do not exist."
Exact(60)
For of non-existent things some exist potentially; but they do not exist, because they do not exist in complete reality.
The existence of conference matrices is not a trivial question, they do not exist for all values of n.
They do not exist".
Officially, they do not exist".
Maybe they do not exist.
The government claims they do not exist.
He contends they do not exist.
We cannot pretend they do not exist.
Yet in finance, they do not exist.
"Off the screen they do not exist as words.
You pretend they do not exist, of course.
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Since I tried Ludwig back in 2017, I have been constantly using it in both editing and translation. Ever since, I suggest it to my translators at ProSciEditing.

Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com