ThmDex – An index of mathematical definitions, results, and conjectures.
Superset of countable union iff superset of every set in the union
Formulation 0
Let $X$ be a D11: Set.
Let $E_n$ be a D11: Set for each $n \in \mathbb{N}$ such that
(i) $\bigcup_{n \in \mathbb{N}} E_n$ is the D77: Set union of $E = \{ E_n \}_{n \in \mathbb{N}}$
Then \begin{equation} \bigcup_{n \in \mathbb{N}} E_n \subseteq X \quad \iff \quad \forall \, n \in \mathbb{N} : E_n \subseteq X \end{equation}
Subresults
R4166: Superset of finite union iff superset of every set in the union
Proofs
Proof 0
Let $X$ be a D11: Set.
Let $E_n$ be a D11: Set for each $n \in \mathbb{N}$ such that
(i) $\bigcup_{n \in \mathbb{N}} E_n$ is the D77: Set union of $E = \{ E_n \}_{n \in \mathbb{N}}$
This result is a particular case of R4164: Superset of union iff superset of every set in the union. $\square$