Dense-in-itself iff Subset of Derivative

Theorem
Let $T$ be a topological space.

Let $A \subseteq T$.

Then:
 * $A$ is dense-in-itself


 * $A \subseteq A'$
 * $A \subseteq A'$

where
 * $A'$ denotes the derivative of $A$.

Proof
$A$ is dense-in-itself

$\iff$ every $x \in A$ is not isolated point in $A$ by definition of dense-in-itself

$\iff$ every $x \in A$ is accumulation point of $A$ by Point is Isolated iff not Accumulation Point

$\iff$ every $x \in A$ belongs to $A'$ by definition of derivative

$\iff$ $A \subseteq A'$ by definition of subset.