Connected and normal Hausdorff with at least two points implies cardinality at least that of the continuum

Statement
Any connected normal Hausdorff space having at least two points is uncountable. In fact, its cardinality is at least equal to the cardinality of the continuum.

Note that the definition of normal space here includes the T1 assumption. The result does not hold for normal-minus-Hausdorff spaces.

Facts used

 * 1) uses::Urysohn's lemma, which, along with the T1 assumption, tells us that normal Hausdorff spaces are Urysohn spaces.
 * 2) uses::Connected and Urysohn with at least two points implies cardinality at least that of the continuum

Proof
The proof follows directly by combining Facts (1) and (2).