First-countable and T1 implies cardinality at most of continuum

This article gives the statement and possibly, proof, of an implication relation between two topological space properties. That is, it states that every topological space satisfying the first topological space property (i.e., firs-countable T1 space) must also satisfy the second topological space property (i.e., topological space with cardinality at most of continuum)
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Statement

Any first-countable space that is also T1 has cardinality at most that of the continuum, i.e. the same as the cardinality of the continuum.

Proof=

Given: A $T_{1}$-space $X$ that is first-countable

To prove: The cardinality of $X$ is at most equal to that of the continuum