Elementary Foundations: An introduction to topics in discrete mathematics

In mathematics we often want to know whether an object with specific desirable properties actually exists. In symbolic language, this is just $(\mathrm{\exists }x)A(x)\text{.}$ Conceptually, this is easy to do: just find an example! (In practice, this can often be quite difficult.)

Once we have found an example for an existential statement, we also often want to know whether there are more examples, or whether the one we have found is unique. Suppose $x_0$ is our concrete example proving $(\mathrm{\exists }x)A(x)\text{.}$ To show that $x_0$ is unique, we should prove the universal statement: $(\mathrm{\forall }y)(A(y)\to (y=x_0))\text{.}$ This translates as the following.

That is, the only way object $y$ can satisfy $A(y)$ is if $y$ is actually our original example $x_0\text{.}$