Correct set notation for "all integers which are not multiples of 7"? What is correct set notation for "all integers which are not multiples of $7$"? My best guess is:

$$ \{ x : (\forall k \in \mathbb{Z})(\neg(7k = x)) \}$$

Or

$$ \{ x : \neg(\exists k \in \mathbb{Z})(7k = x) \}$$

However this seems unlike other examples I have seen.

Is there are proper way to denote this set in set notation?
I'd generally go with $\{x\in\mathbb{Z}: 7 \nmid x\}$, though this does presuppose familiarity with the $\mid \, \nmid$ notation.
What about:
{7j+k (∀j∈Z) (∀k∈{1,2,3,4,5,6})}
?