Cardinal Number Less than Ordinal/Corollary
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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: $\S 10.13$