Existence and uniqueness of an abstract mathematical problem Suppose P is an abstract mathematical problem, an element $x$ is a solution of P if P(x) is true. The uniqueness is defined as P is unique determined if any two solution of P is equal. 
Question: I have read that for the definition of uniqueness one does not require the solution exist. However, how can this be consistent as if there exists no solution, how can I check if any two solution is equal? 
 A: Uniqueness means:
$$\forall x,y\ \ P(x)\land P(y)\implies x=y$$
If there is no solution, both $P(x)$ and $P(y)$ are false, which makes the implication true (vacuous truth concept).
The nice thing is that the opposite:
$$\forall x,y\ \ P(x)\land P(y)\implies x\ne y$$
is also true. 
In other words, anything you would say about things that do not exist is true (you could say "My car is a Ferarri", and that would be true if you don't have a car).
A: Non-uniquenes of solution means that there are at least two solutions. But there is not even one. In other words:

Suppose that $P(x)$ is true, $P(y)$ is true and $x\neq y$. Since $P$
  has no solution, $P(x)$ is false. Contradiction.

A: This general Uniqueness Theorem article is helpful.
It is stated there that
"in such cases, a uniqueness theorem is normally combined with an existence theorem"
The antiderivative of any function is unique, up to a constant. But they might not exist for a wild and crazy function.
Following is a list uniqueness theorems in the mathematical literature without a corresponding existence theorem/statement/boundary condition. I encourage anyone with some interesting examples to edit this answer and add to this (short?) list.

1.
