Cardinal Number Less than Ordinal

Theorem

Let $S$ be a set.

Let $\card S$ denote the cardinal number of $S$.

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

Then:

$\card S \le x$

Corollary

Let $x$ be an ordinal.

Let $\card x$ denote the cardinal number of $x$.

Then:

$\card x \le x$

Proof

Since $S \sim x$, it follows that:

$x \in \set {y \in \On : S \sim y}$

By Intersection is Subset: General Result, it follows that:

$\ds \bigcap \set {y \in \On: S \sim y} \subseteq x$

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Therefore $\card S \le x$ by the definition of cardinal number.

$\blacksquare$

Sources

• 1971: Gaisi Takeuti and Wilson M. Zaring: Introduction to Axiomatic Set Theory: $\S 10.12$