It’s not hard to check that the subobject classifier $\Omega$ is trivially fibrant, and so its homotopy type is trivial. Orthogonality of $\Omega \to 1$ against a map $i : A \to B$ corresponds to the property that any subobject $A' \to A$ extends to a subobject $B' \to B$ such that $i^*B' = A'$; when $i$ is mono, this is always possible, with canonical solutions given by the direct image $\exists_i A'$ and the dual image $\forall_i A'$.

Nothing here is special to simplicial sets: this argument anpplies equally in any model structure on a topos in which all cofibrations are monomorphisms.

This argument appears (tersely!) in the proof of Theorem  1.4.3 of Cisinski 2006, [*Les préfaisceaux comme modèles des types d’homotopie (Presheaves as models for homotopy types)*, 2006](http://www.numdam.org/item/AST_2006__308__R1_0/) (thanks to Tim Campion in comments for the precise reference).  A couple of closely related arguments — firstly the fibrancy of a universe classifying certain fibrations, corresponding to showing that those fibrations extend along trivial cofibrations, and secondly the univalence of this universe — appear in my 2012 paper with Chris Kapulkin, [*The simplicial model of univalent foundations (after Voevodsky)*](https://arxiv.org/abs/1211.2851), in Sections 2.1, 2.2, and 3.2.