Non-Finite Cardinal is equal to Cardinal Product/Corollary

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Corollary to Non-Finite Cardinal is equal to Cardinal Product

Let $S$ be a set that is equinumerous to its cardinal number.

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

Let:

$\card S \ge \omega$

where $\omega$ denotes the minimally inductive set.


Then:

$\card {S \times S} = \card S$


Proof

By hypothesis:

$S \sim \card S$


By Cartesian Product Preserves Cardinality:

$S \times S \sim \card S \times \card S$

Therefore:

\(\ds \card {S \times S}\) \(=\) \(\ds \card {\card S \times \card S}\) Equivalent Sets have Equal Cardinal Numbers
\(\ds \) \(=\) \(\ds \card S\) Non-Finite Cardinal is equal to Cardinal Product

$\blacksquare$


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