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Would any Swedish speaker like to help me with the last sentence? I'm not sure if it's complete. It looks like it means "but you don't...".

Meaning is: "without you (there's) no one".

I'm not sure if I understand.
Does that mean something like "without you no one can live" or "without you no one can exist"?

Hard to tell since the sentence is short, but I think you can say: "without you (there is) no one (for me)."

Does it make it clearer?

Hun skrev "Ali ti ne..." och det betyder inte "utan dig ingen"

"Volim te najviše na celom svetu. Ti si ta koja me čini živim. Bez tebe ništa". Sista delen stämmer inte med svenska texten.
"Ali ti ne..." betyder "men du inte..."

Pia,
I wrote "without you there is no one other for me" (there is no one else with whom I would be if I weren't with you). I couldn't write nothing shorter cause it wouldn't be in the spirit of Serbian language.
Do you think it's ok?

Hello Maki,
I think that is exactly the meaning for it, (what you wrote) even if it is written very short in Swedish. "without you, no one". Hope it helps.

It helped a lot!
Thank you Pia!

You are welcome Maki, I'm glad to help