Equivalence of Definitions of Cover of Set

Theorem
Let $S$ be a set.

Definition 1 Implies Definition 2
Let $\CC$ be a set of sets such that:
 * $\ds S \subseteq \bigcup \CC$

where $\bigcup \CC$ denotes the union of $\CC$.

By definition of subset:
 * $\forall s \in S : s \in \ds \bigcup \CC$

By definition of set union:
 * $\forall s \in S : \exists C \in \CC : s \in C$

Definition 2 Implies Definition 1
Let $\CC$ be a set of sets such that:
 * $\forall s \in S : \exists C \in \CC : s \in C$

From Set is Subset of Union (General Result):
 * $\forall C \in \CC : C \subseteq \ds \bigcup \CC$

By definition of subset:
 * $\forall s \in S : s \in \ds \bigcup \CC$

By definition of subset:
 * $S \subseteq \ds \bigcup \CC$