Cardinal Number Less than Ordinal/Corollary

Corollary to Cardinal Number Less than Ordinal

Let $x$ be an ordinal.

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

Then:

$\card x \le x$

Proof

By Set Equivalence behaves like Equivalence Relation:

$x \sim x$

By Cardinal Number Less than Ordinal:

$\card x \le x$

$\blacksquare$

Sources

• 1971: Gaisi Takeuti and Wilson M. Zaring: Introduction to Axiomatic Set Theory: $\S 10.13$