# Injection implies Cardinal Inequality

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## Theorem

Let $S$ and $T$ be sets.

Let $f: S \to T$ be an injection.

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

Let:

- $T \sim \card T$

where $\sim$ denotes set equivalence

Then:

- $\card S \le \card T$

## Proof

Let $f \sqbrk S$ denote the image of $S$ under $f$.

\(\displaystyle S\) | \(\sim\) | \(\displaystyle f \sqbrk S\) | Set is Equivalent to Image under Injection | ||||||||||

\(\displaystyle \) | \(\subseteq\) | \(\displaystyle T\) | Image Preserves Subsets | ||||||||||

\(\displaystyle \leadsto \ \ \) | \(\displaystyle \card S\) | \(=\) | \(\displaystyle \card {f \sqbrk S}\) | Equivalent Sets have Equal Cardinal Numbers | |||||||||

\(\displaystyle \) | \(\le\) | \(\displaystyle \card T\) | Subset implies Cardinal Inequality |

$\blacksquare$