Subset of Ordinal implies Cardinal Inequality

Theorem

Let $S$ be a set.

Let $x$ be an ordinal such that $S \subseteq x$.

Then:

$\card S \le \card x$

where $\card S$ denotes the cardinality of $S$.

Proof

Since $x$ is an ordinal, it follows that $x \sim \card x$ by Ordinal Number Equivalent to Cardinal Number.

This satisfies the hypothesis for Subset implies Cardinal Inequality.

Therefore:

$\card S \le \card x$

$\blacksquare$

Sources

• 1971: Gaisi Takeuti and Wilson M. Zaring: Introduction to Axiomatic Set Theory: $\S 10.23$