Banned. “There exists” works exactly like the more common “there is.” Although we use “it” as a subject for verbs describing the weather (“it is raining”; “it is cold”), we usually use “there is” as an introductory statement, and never “it is.”

Does there exist or exist?

The grammatical subject is “there”, which is singular, and the traditional rule says that the verb should be singlar “exists” and “does”; but many people in sentences like this make the verb agree with the logical subject, and say “exist” and “do”.

What does there exist mean?

Usage. The ∃ (there exists) symbol is used in math to express the existence of a variable. For example, the symbol is usually used in an expression like this. ∃x∈N:x is even. In plain language, this expression means there exists a variable x belonging to the set of natural numbers such that x is even.