ThmDex – An index of mathematical definitions, results, and conjectures.
Result R4143 on D77: Set union
Countable union is an upper bound to each set in the union
Formulation 0
Let $E_n$ be a D11: Set for each $n \in \mathbb{N}$ such that
(i) $\bigcup_{n \in \mathbb{N}} E_n$ is a D77: Set union for $\{ E_n \}_{n \in \mathbb{N}}$
Then \begin{equation} \forall \, k \in \mathbb{N} : E_k \subseteq \bigcup_{n \in \mathbb{N}} E_n \end{equation}
Formulation 1
Let $E_n$ be a D11: Set for each $n \in \mathbb{N}$ such that
(i) $\bigcup_{n \in \mathbb{N}} E_n$ is a D77: Set union for $\{ E_n \}_{n \in \mathbb{N}}$
Then \begin{equation} E_1 \subseteq \bigcup_{n \in \mathbb{N}} E_n, \quad E_2 \subseteq \bigcup_{n \in \mathbb{N}} E_n, \quad E_3 \subseteq \bigcup_{n \in \mathbb{N}} E_n, \quad \ldots \end{equation}
Proofs
Proof 0
Let $E_n$ be a D11: Set for each $n \in \mathbb{N}$ such that
(i) $\bigcup_{n \in \mathbb{N}} E_n$ is a D77: Set union for $\{ E_n \}_{n \in \mathbb{N}}$
This result is a particular case of R4142: Union is an upper bound to each set in the union. $\square$