Uniqueness Quantifier

Sometimes you want to say “there’s exactly one.”

$\exists! x \; P(x)$

“There exists a unique x such that P(x).”

Example:

$\exists! x \; (x + 3 = 5)$ — “There’s exactly one number that, added to 3, gives 5”
That number is 2. Only 2. No other.

What it Really Means

$\exists! x \; P(x)$ is shorthand for:

$\exists x \; (P(x) \land \forall y \; (P(y) \to y = x))$

Let’s break this down:

Part 1: $\exists x \; P(x)$ — “At least one exists”

Without this, there might be zero. We need at least one.

Part 2: $\forall y \; (P(y) \to y = x)$ — “No duplicates”

If any other y also satisfies P, then y must be the same as x. This rules out having two different things that work.

Together: “There’s one, and anything else that works is actually the same one.”

One exists, and it’s the only one.