Learn Hardcore Dutch: Die fiets is nieuw, maar de mijne is oud.

die is a demonstrative pronoun equivalent to that in English. It points out a specific bike: Die fiets = That bike. In contrast, de on its own is the definite article the, so de fiets would just mean the bike without emphasizing which one.

What’s the difference between die and deze?

Both are demonstratives:

deze = this (for something close in space, time or context)
die = that (for something further away)
Example: Holding a bike: Deze fiets is nieuw. Pointing at a bike across the room: Die fiets is nieuw.

Why are nieuw and oud not inflected with -e (as in nieuwe or oude)?

Dutch adjectives get -e when they’re attributive (directly before a noun): de nieuwe fiets, een oude auto. But when they come after a verb like zijn (to be), they’re predicative and stay in their base form:
Die fiets is nieuw, maar de mijne is oud.

What does de mijne mean? Why not just mijn or mijne?

mijne is the strong (noun-replacing) form of mine. You use mijne when you drop the noun (fiets) and want to say mine. mijn is a possessive adjective and needs a noun after it: mijn fiets = my bike.

Why is there still an article de before mijne?

Because strong possessive pronouns in Dutch require the corresponding definite article. Since fiets is a de-word, you say de mijne = the mine = mine. If it were a het-word, you’d use het mijne.

Can I say maar mijn fiets is oud instead of maar de mijne is oud?

Yes.

Maar mijn fiets is oud literally repeats fiets: But my bike is old.
Maar de mijne is oud drops fiets and replaces it with de mijne: But mine is old.
Both are correct; the second avoids repeating the noun.

Why is the verb is in second position in the second clause after maar?

Dutch main clauses follow the verb-second (V2) rule. Even after a coordinating conjunction like maar, the finite verb stays in second position:
1) Maar (conjunction)
2) de mijne (subject)
3) is (verb)
4) oud.