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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.

Read the initial universal quantifiers as governing the conjunction of irreflexivity and transitivity. Explicitly, the intended first-order sentence assertsThus $Q$ is an irreflexive relation, a transitive relation, and a serial relation.

Suppose a first-order model had a finite nonempty domain of cardinality $N$. Choose $a_0$. Seriality supplies successive elements $a_1,\dots ,a_N$ with $Q(a_i,a_{i+1})$. By the pigeonhole principle, $a_i=a_j$ for some $i<j$. Repeated transitivity along that part of the chain gives $Q(a_i,a_j)=Q(a_i,a_i)$, contradicting irreflexivity. Hence there is no finite nonempty model. This is the infinite-model property of a serial strict partial order.

An infinite model exists: take domain $\mathbb{N}$ and $Q(x,y)$ to mean $x<y$. It is irreflexive and transitive, and $y=x+1$ witnesses seriality for every $x$.

The quantifier-scope convention matters: giving each printed universal quantifier only the immediately following atomic scope would leave the transitivity implication with free variables and would not express the claimed theorem. The explicit parenthesisation above records the reading under which the source's requested assertion is valid.

A binary relation is serial when every domain element has at least one outgoing related element. A serial relation may have cycles; combining seriality with an irreflexive relation and a transitive relation instead forces a nonempty domain to be infinite, by the infinite-model property of a serial strict partial order.