Infinite-model property of a serial strict partial order

A strict partial order that is a serial relation has no finite nonempty first-order model. If the domain had $N$ elements, seriality would produce a chain of $N+1$ successive elements. The pigeonhole principle supplies a repeated element, and transitivity along the resulting cycle contradicts irreflexivity. An infinite example is the usual strict order on the natural numbers: every element has its successor. The nonempty-domain convention is essential; the displayed universal conditions hold vacuously on an empty domain if such domains are permitted.