If you have a child who is worried that Santa Claus might not exist, you can reassure them using this proof.

Call this sentence P:

If this sentence is true, Santa Claus exists.

Suppose P is true. P says “If P is true, Santa Claus exists.” So, Santa Claus exists.

So, we've proved that if P is true, Santa Claus exists.

But that's just what P says!

So, P is true.

So, Santa Claus exists!

Merry Christmas!

@johncarlosbaez
Define "Santa Claus" as the coolest entity having all of the attributes normally attributed (white beard, list making, gift giving, etc).

Clearly something that exists is cooler than something that doesn't exist.

Therefore, Santa Claus exists!