I'm looking at Euclid's Theorem (the infinitude of primes).
The standard proof assumes there are finitely many primes (and proceeds to contradiction). It involves $P :=$ the product of all the primes, and $Q := P+1$.
Since every prime divides $P$, no prime can divide $Q$; since if something divides two numbers it must also divide their difference, and the difference between $P$ and $Q$ is $1$, and nothing divides $1$ except $1$.
But... why does nothing divide $1$ except $1$? Given the definition of divisibility
$$a\mid b \iff \exists x : ax = b$$
how can you prove that $1$ has only one divisor?