Set Less than Cardinal Product

Theorem

Let $S$ and $T$ be sets.

Let $T$ be nonempty.

Suppose that $S \times T \sim \card {S \times T}$.

Then:

$\card S \le \card {S \times T}$

Proof

Let $y \in T$.

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

$\map f x = \tuple {x, y}$

If $\map f {x_1} = \map f {x_2}$, then $\tuple {x_1, y} = \tuple {x_2, y}$ 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 $\card S \le \card {S \times T}$

$\blacksquare$

Sources

• 1971: Gaisi Takeuti and Wilson M. Zaring: Introduction to Axiomatic Set Theory: $\S 10.25$