Acchaa, language origin is mysterious, but as far as I’ve heard, ’na’ is the original Sanskrit word, and ’nahiin’ developed in Hindi, possibly as a combination of ’na’ plus another word. 
I think by itself, it is like an informal ’No’ (Naw) instead of ’(Ji) Nahiin”. But ’na’ also can be used like ”don’t” or ”neither”; 
”Dhobi ka kutta na ghar ka na ghaat ka” The washerman’s dog belongs neither at home nor on the river bank. 

Why is it because the dog is dirty? Poor thing would have nowhere to go would it?

I saw the ”he” part somewhere on its own- in fact this is what made me realize that Nahi was a portmanteau. The context was it meant ”is” but a different kind of is than है. Wishing I had documented this sighting…. but like birdwatching I’m sure to encounter this elusive species again. (it was somewhere around here actually)

P.S. ’na’- (&’an’-) can prefix words, to make something be ”un-something”: pasand, napasand, etc.

ही is at the end of the 9th ”Glossaries Alive” !

Sentences with Nahin or Na. 9example i.e उदाहरण

1.मैं तुमसे बात नहीं करना चाहता | Main tumesy baat nahin karna chahta. Baat is also called. वार्तालाप Vaartalaap i.e conversation means baatchee

You can add Huun or without that also it is spoken.

2 ..न तुम मुझे याद करते और न मैं आता | Na tum mujhe yad kartey aur na main aata. had you not remembered me, I would not havet come.

3. मेरे न कहने पर भी वह नहीं माना | Mere na kahney par bhee vah nahin maana. Despite my telling NO, he did not listen to.

4.मुझे खीर नहीं चाहिए | Mujhe kheer nahin chahiye. I do not want sweet pudding.
Kheer is also called Payas (Payasam in South India).

Mat मत means also NO. मुझे तंग मत करो Mujhe tang mat karo. Do not pester .
me.

Please note मतदान matdaan means casting of votes.
एकमत means all agree,

For information Kheer is a wonderful sweet dish( rice or sago or jackfruit,, almond. For this , milk is must+ sugar. In south Jaggery is also used while making payasam.

tks.

Regards

suresh