Set Less than Cardinal Product

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Theorem

Let $S$ and $T$ be sets.

Let $T$ be nonempty.

Suppose that $S \times T \sim \left|{ S \times T }\right|$.


Then:

$\left|{ S }\right| \le \left|{ S \times T }\right|$


Proof

Let $y \in T$.


Define the mapping $f : S \to S \times T$ as follows:

$f\left({x}\right) = \left({ x,y }\right)$

If $f\left({x_1}\right) = f\left({x_2}\right)$, then $\left({ x_1 , y }\right) = \left({ x_2 , y }\right)$ by the definition of $f$.

It follows that $x_1 = x_2$ by Equality of Ordered Pairs.

Thus, $f : S \to S \times T$ is an injection.


By Injection implies Cardinal Inequality, it follows that $\left|{S}\right| \le \left|{S \times T}\right|$

$\blacksquare$


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