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'were non-existent' is correct and can be used in written English.
It is an example of the past tense of "to be". You can use it when you want to describe something that did not exist in the past. For example: Before the invention of the printing press, books were non-existent.
Dictionary
were non-existent
adjective
Alternative spelling of nonexistent
Exact(60)
In those days, human rights were non-existent, harsh military punishment existed in place of justice and sex trafficking was an all-too-common practice.
Details were non-existent, however.
"I felt my employee rights were non-existent.
The ethical concerns were non-existent, said Anand.
Mobile phones, internet and e-mail were non-existent.
Companies were non-existent and this meant there were no bosses.
Shareholder class actions, which were non-existent in Australia until the 1990s, offer another enforcement route.
Building an audience proved to be quite costly, while advertising revenues were non-existent.
WOMEN'S RIGHTS At the time of the 2001 invasion, women's rights were non-existent.
The aggression and hostility he used took England's tail out of the game – they were non-existent as a batting force.
Big Star had always been a studio band rather than a touring band, but by the time Third was finished, they were non-existent.
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Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com