No Arithmetic Sequence of 4 Primes with Common Difference 2/Corollary

Definition
Let $n \in \Z$.

Let $S_k \left({n}\right) = \left\{{n, n+2, n+4, \ldots, n+2k}\right\}$ where $k > 2$.

Then it can not be the case that all elements of $S$ are primes.

Proof
From Prime Triplet Unique the set $S_2 \left({3}\right) = \left\{{3, 5, 7}\right\}$ consists only of primes.

From No Prime Quadruplets, the set $S_3 \left({n}\right)$ contains at least one non-prime.

As $S_{k-1} \left({n}\right) \subset S_k \left({n}\right)$ it follows that for $k \ge 4$, the set $S_k \left({n}\right)$ also contains at least one non-prime.