No Arithmetic Sequence of 4 Primes with Common Difference 2

Definition
There exist no prime quadruplets.

That is, there exists no $n \in \Z_{>0}$ such that $n, n+2, n+4, n+6$ are all prime.

Proof
Any prime quadruplet must contain as a subset a prime triplet.

From Prime Triplet Unique, the only one of these is $\left\{{3, 5, 7}\right\}$.

The only sets of the form $\left\{{n, n+2, n+4, n+6}\right\}$ containing $\left\{{3, 5, 7}\right\}$ are:


 * $(1): \quad \left\{{1, 3, 5, 7}\right\}$: as $1$ is by convention not a prime, then this is not a prime quadruplet.


 * $(2): \quad \left\{{3, 5, 7, 9}\right\}$: as $9 = 3 \times 3$ is not a prime, then this is not a prime quadruplet.

There are no more possible $\left\{{n, n+2, n+4, n+6}\right\}$ all prime.