Learn Farsi: در بسته نیست، باز است. - The door is not closed; it is open.

nist is the negative form of the verb "to be" for the third-person singular ("he/she/it"). Because the subject of the sentence is dar (the door), which is an "it", we use nist to mean "is not". If you were saying "I am not", you would use nistam.

What is the relationship between nist and ast in this sentence?

They are the negative and positive versions of the exact same verb form! ast means "is" (for he/she/it), and nist means "is not". This sentence perfectly contrasts the two to describe the door: "is not" (nist) closed, "is" (ast) open.

Why is there no Farsi word for "it" before bāz ast?

In Farsi, you can drop subject pronouns like "it" (ān) when the context makes it obvious. Since the first half of the sentence already established that we are talking about the door, we don't need to say "it" again. The verb ast already contains the meaning of "is".

How would a native speaker say bāz ast in casual, everyday speech?

In informal spoken Farsi, the word ast (is) often attaches to the end of the previous word and changes into a short "e" sound. So, instead of carefully pronouncing bāz ast, a native speaker will usually say bāz-e. However, the negative word nist generally stays exactly the same in spoken Farsi.