Injection implies Cardinal Inequality

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

$\blacksquare$