- '-지 않다' is an auxiliary verb [보조 용언] which is used to negate a verb or adjective. For example:
(1) 나는 여자친구가 있어요. I have a girl friend.
(2) 나는 여자친구가 없어요. I don't have a girlfriend. (I have no girl friend.)
(3) 나는 여자친구가 있지(는) / 있진 (contraction of 있지는) 않아요. I don't have a girlfriend.
(4) 나는 여자친구가 없지(는) / 없진 (contraction of 없지는) 않아요. I have a girlfriend.
No. 3 and No. 4 are more emphatic than No. 1 and No. 2, but they are not as broadly used.
Adding the particle '-는' is also optional and it is used for emphasis.
- '안 없어요?' sounds like a Korean language beginner talking. It will never be used by a native Korean speaker unless (s)he wants to make a joke.
"여자친구 안 없어요?" sounds like asking if you have a girl friend, but it is not idiomatic. However, "여자친구 안 있어요?" would be understood to mean "Don't you have a girl friend?".
'안' (네이버 국어사전 link) shows that it is an adverb (부사) which is a contracted form of '아니' and it is used to negate a verb or adjective or to make an opposite meaning (of an adjective, e.g., 덥다 (hot) vs 안 덥다 (not hot)).
‘아니1(1. 부정이나 반대의 뜻을 나타내는 말)’의 준말.
안 벌고 안 쓰다. (Literally) Subject doesn't earn money and doesn't spend it.
안 춥다. Not cold. (춥다, cold)
비가 안 온다. (Literally) The rain doesn't come. (It is not raining.)