Nem azért szigetelünk, mert hideg van, hanem azért, mert takarékoskodunk.

No. Here nem is not really denying the action szigetelünk; it is denying the reason introduced by azért.

So the meaning is:

It is not for that reason that we are insulating...

not

We are not insulating...

If you wanted to say We are not insulating, that would simply be Nem szigetelünk.

This is a very common Hungarian pattern for rejecting one reason and giving the real one.

It corresponds to English:

not because X, but because Y

In this sentence:

• Nem azért ... mert hideg van = not because it is cold
• hanem azért, mert takarékoskodunk = but because we are economizing / trying to save

So the structure is very idiomatic and useful to learn as a chunk.

Because Hungarian is setting up a neat contrast:

• first azért = the reason being rejected
• second azért = the real reason being given

It makes the sentence balanced and explicit.

You may also hear a slightly shorter version:

Nem azért szigetelünk, mert hideg van, hanem mert takarékoskodunk.

That is also natural. But the version with both azért sounds especially clear and symmetrical.

Here azért means something like for that reason. It points forward to the clause introduced by mert.

So:

• azért ..., mert ... = for the reason that ... / because ...

Be aware that azért can have slightly different uses in other contexts, sometimes closer to that’s why / therefore / for it. But in this sentence it is part of the common pattern azért ..., mert ....

Because hanem is the normal word for but rather after a negative statement.

Hungarian usually uses:

• de for general contrast: but
• hanem when the first idea is negated and the second replaces or corrects it: not X, but Y

This sentence is exactly that kind of correction:

not because it is cold, but because we are economizing

So hanem is the right choice.

Because Hungarian normally expresses this idea with van:

• hideg van = it is cold

Hungarian does not use a dummy subject like English it. So instead of it is cold, Hungarian simply says cold is.

In very short, informal reactions, Hideg! can work, but in a full clause like this, hideg van is the normal form.

Hungarian often leaves the object unstated when it is obvious from context or when the speaker is talking about the activity in general.

So szigetelünk can mean:

• we are insulating
• we are doing insulation work

If you wanted to name the object, you could say something like:

A házat szigeteljük = We are insulating the house

So the lack of an object here is completely normal.

They mark first person plural in the present tense:

• szigetelünk = we insulate / we are insulating
• takarékoskodunk = we economize / we are economizing

That is why Hungarian does not need a separate subject pronoun mi for we here. The verb ending already tells you who the subject is.

The difference between -ünk and -unk is due to vowel harmony.

It means we are being economical, we are economizing, or we are trying to save.

In this context, it probably suggests saving:

• energy
• heating costs
• money in general

So it is not necessarily about saving money in a bank account. It is more about living or acting in a cost-conscious, resource-conscious way.

Because each half of the contrast has its own reason clause:

• mert hideg van
• mert takarékoskodunk

Repeating mert makes the structure clear and parallel. Hungarian strongly prefers that here.

It is similar to English, where we also normally say:

not because X, but because Y

Because Hungarian normally separates these clauses with commas:

• before mert
• before hanem
• and again before the second mert

So the punctuation here is standard Hungarian clause punctuation, not something unusual or optional.

Yes. The beginning Nem azért puts the focus on the reason, not on the action.

The point is:

We do insulate — just not for that reason.

That is why the sentence starts with Nem azért. It immediately tells the listener that a reason is about to be rejected and replaced with the real one.