Cardinal of Union Equal to Maximum

Theorem

Let $S$ and $T$ be sets that are equinumerous to their cardinal number.

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

Suppose $S$ is infinite.

Then:

$\card {S \cup T} = \map \max {\card S, \card T}$

Proof

Let $x$ denote $\map \max {\card S, \card T}$.

$x = \card S$ if $\card T \le \card S$

$x = \card T$ if $\card S \le \card T$

$x = \card S$ or $x = \card T$

In either case, it follows by Subset of Union that:

$x \le \card {S \cup T}$

$\Box$

If $\card T = 1$ or $\card T = 0$, it follows that $\card {S \cup T} = \card S = x$ by the definition of an infinite set.

If $1 < \card T$, then $\card {S \cup T} \le \card {S \times T}$ by Cardinal of Union Less than Cardinal of Cartesian Product.

But by Cardinal Product Equal to Maximum, it follows that $\card {S \cup T} \le x$.

$\Box$

Combining the two lemmas, it follows that $x = \card {S \cup T}$

$\blacksquare$