In a model category, when weak equivalences are inverted, nothing else gets inverted. It follows that weak equivalences satsify 2-out-of-6. But the first sentence takes some work to show. Is there a more direct way to demonstrate 2-out-of-6?
For reference, 2-out-of-6 says that if $v \circ u$ and $w\circ v$ are weak equivalences, then $w\circ v \circ u$ is a weak equivalence (in light of 2-out-of-3, the conclusion is equivalent to saying that any of $u,v,w$ is a weak equivalence). This property apparently features in the definition of a homotopical category. It follows from the property above because isomorphisms satisfy 2-out-of-6, so the preimages of isomorphisms under a functor also satsify 2-out-of-6.
I'm interested in having a direct argument because 2-out-of-6 leads to a simple proof that if $f$ is a homotopy equivalence (i.e. there exists $g$ such that $gf$ and $fg$ are both either left or right homotopic to identity maps) then $f$ is a weak equivalence. I would expect this fact to have a simple proof, but in fact Dwyer-Spalinski and Hovey at least only prove this directly with some fibrancy/cofibrancy restrictions on the objects involved. The proof using 2-out-of-6 goes like this: it's easy to see that if $h$ is left or right homotopic to $k$, then if one is a weak equivalence then so is the other; so if $f$ has homotopy inverse, then $fg$ and $gf$ are both weak equivalences, so 2-out-of-6 implies that $f$ and $g$ are each weak equivalences.