Cardinality of Basis of Sorgenfrey Line not greater than Continuum

Theorem
Let $T = \struct {\R, \tau}$ be the Sorgenfrey line.

Let
 * $\BB = \set {\hointr x y: x, y \in \R \land x < y}$

be the basis of $T$.

Then $\card \BB \le \mathfrak c$

where
 * $\card \BB$ denotes the cardinality of $\BB$
 * $\mathfrak c = \card \R$ denotes the continuum.

Proof
Define a mapping $f: \BB \to \R \times \R$:
 * $\forall I \in \BB: \map f I = \tuple {\min I, \sup I}$

That is:
 * $\map f {\hointr x y} = \tuple {x, y} \forall x, y \in \R: x < y$

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

Let $I_1, I_2 \in \BB$ such that:
 * $\map f {I_1} = \map f {I_2}$

So:
 * $I_1 = I_2$

Thus $f$ is an injection.

By Injection implies Cardinal Inequality:
 * $\card \BB \le \card {\R \times \R}$

By Cardinal Product Equal to Maximum:
 * $\card {\R \times \R} = \map \max {\mathfrak c, \mathfrak c}$

Thus:
 * $\card \BB \le \mathfrak c$