Dominates is an Ordering

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Theorem

Let $S$ and $T$ be cardinals.

Let $S \preccurlyeq T$ denote that $S$ is dominated by $T$.

Let $\mathbb S$ be any set of cardinals.

Then the relational structure $\left({\mathbb S, \preccurlyeq}\right)$ is a poset.

That is, $\preccurlyeq$ is an ordering (at least partial) on $\mathbb S$.

From the definition, a cardinal is a set, so standard set theoretic results apply.

So, checking in turn each of the criteria for an ordering:

For any cardinal $S$, the identity mapping $I_S: S \to S$ is a is an injection, so $\forall S \in \mathbb S: S \preccurlyeq S$.

So $\preccurlyeq$ is reflexive.

$\Box$

Let $S_1, S_2, S_3 \in \mathbb S$ such that $f: S_1 \to S_2$ and $g: S_2 \to S_3$ be injections.

Then from Composite of Injections is an Injection, $g \circ f$ is an injection and so $S_1 \preccurlyeq S_3$.

So $\preccurlyeq$ is transitive.

$\Box$

Suppose $S \preccurlyeq T$ and $T \preccurlyeq S$.

Then from the Cantor-Bernstein-Schroeder Theorem, $S \sim T$ and, as $S$ and $T$ are cardinals, $S = T$ by definition.

So $\preccurlyeq$ is antisymmetric on a set of cardinals.

$\Box$

Hence the result.

$\blacksquare$

Steven A. Gaal: Point Set Topology (1964)... (previous)... (next): Introduction to Set Theory: $2$. Set Theoretical Equivalence and Denumerability
T.S. Blyth: Set Theory and Abstract Algebra (1975): $\S 8$: Theorem $8.2$ (Corollary)