Translation:The city's library is open, but the cinema is closed.
I would guess that "The city's library..." is also an appropriate translation?
Why is "van" in this sentence? Couldn't you say, "A könyvtár nyitva" for "the library is open"? And if so, then I don't see why "van" is necessary here.
Coloquially, just using nyitva seems to be okay, but the problem here is that it's not an adjective.
The -va/-ve form is a special suffix that transforms verbs (nyit - to open) into adverbs. Nyitva is called an "adverbial participle", and as such it needs a verb to attach to.