Cardinality of Basis of Sorgenfrey Line not greater than Continuum

Theorem
Let $T = \left({\R, \tau}\right)$ be the Sorgenfrey Line.

Let
 * $\mathcal B = \left\{{\left[{x \,.\,.\, y}\right): x, y \in \R \land x < y}\right\}$

be the basis of $T$.

Then $\left\vert{\mathcal B}\right\vert \leq \mathfrak c$

where
 * $\left\vert{\mathcal B}\right\vert$ denotes the cardinality of $\mathcal B$,
 * $\mathfrak c = \left\vert{\R}\right\vert$ denotes continuum.

Proof
Define a mapping $f: \mathcal B \to \R \times \R$:
 * $\forall I \in \mathcal B: f \left({I}\right) = \left({\min I, \sup I}\right)$.

That is
 * $f \left({\left[{x \,.\,.\, y}\right)}\right) = \left({x, y}\right) \forall x, y \in \R: x < y$.

We will show that $f$ is an injection dy definition.

Let $I_1, I_2 \in \mathcal B$ such that
 * $f \left({I_1}\right) = f \left({I_2}\right)$.

So:
 * $I_1 = I_2$.

Thus $f$ is an injection.

By Injection implies Cardinal Inequality:
 * $\left\vert{\mathcal B}\right\vert \leq \left\vert{\R \times \R}\right\vert$.

By Cardinal Product Equal to Maximum
 * $\left\vert{\R \times \R}\right\vert = \max\left({\mathfrak c, \mathfrak c}\right)$

Thus
 * $\left\vert{\mathcal B}\right\vert \leq \mathfrak c$.