If a single point is added or removed from a poset, its dimension stays the same or changes by $1$. How to prove it?
I heard that it could be proven using Łoś's theorem, but, despite trying, I can't obtain a proof from this hint (nor any other proof).
My motivation is mainly to understand how Łoś's theorem works by looking at its application to a problem which isn't very general, yet non-trivial. (Other answers, not relying on Łoś's theorem, are welcome too, though, enabling comparison possibly giving better understanding of what the application of Łoś's theorem actually contributes here.)