We choose our category of spaces to be compactly generated weak Hausdorff spaces for convenience, denoted $CGWH$.

Questions:

1.) Is there any sort of slick argument to verify that CGWH with the Quillen model structure is a right-proper (closed) model category?

2.) If we give the following presentation of the model structure on SSet:

Cofibrations are monomorphisms

Fibrations have the RLP with respect to all horn inclusions $\Lambda^n_i \subseteq \Delta^n$ for $0\leq i \leq n$.

Or instead of the characterization of cofibrations, we could instead give:

Trivial fibrations have the RLP with respect to all inclusions of the boundary $\partial^n \subseteq \Delta^n$.

(The point of picking a nice presentation is that the (morally) right choice of definition often simplifies a proof.)

Is there any way to verify the model category axioms more easily? The proofs I've seen appeal to all of the hard work done in question 1. It seems like one should be able to verify the axioms for SSet more easily than the case of CGWH spaces.