Hey Anti-Theists! Prove Your Claim

[Mister Agenda]

*I misunderstood the usage of the term anti-theist in the OP but that mistake was corrected by the posts of others.

Not a misuse Chad. The definition of anti-theist that you gave is a valid one. The term 'gnostic atheist' is actually less accurate, etymologically, but it is in common use and has a well accepted definition that meets the needs of the OP.

The definition of anti-theist is in common use, as well:

anti theist
Web definitions
Antitheism is active opposition to theism. The term has had a range of applications; in secular contexts, it typically refers to direct opposition to organized religion or to the belief in any deity, while in a theistic context, it sometimes refers to opposition to a specific god or gods.

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I just don't get the mindset that produces a statement along the lines of "I am sure that my god exists and the burden of proof lies upon anyone who doubts me".

Is it any different than saying I dismiss all existing evidence of God's existence and the burden of proof lies upon the theist to provide new evidence I can't possibly dispute?

Yes. Despite deliberately wording it to make it appear that the speaker didn't carefully consider the evidence, the latter assigns the burdern of proof correctly, on the person trying to establish the existence of something.