ThmDex – An index of mathematical definitions, results, and conjectures.
Result R1034 on D80: Power set
Power set is closed under unions
Formulation 0
Let $X$ be a D11: Set.
Let $E_j$ be a D11: Set for each $j \in J$.
Then \begin{equation} \forall \, j \in J : E_j \in \mathcal{P}(X) \quad \Rightarrow \quad \bigcup_{j \in J} E_j \in \mathcal{P}(X) \end{equation}
Formulation 1
Let $X$ be a D11: Set.
Let $E_j$ be a D11: Set for each $j \in J$.
Then \begin{equation} \forall \, j \in J : E_j \subseteq X \quad \Rightarrow \quad \bigcup_{j \in J} E_j \subseteq X \end{equation}
Proofs
Proof 0
Let $X$ be a D11: Set.
Let $E_j$ be a D11: Set for each $j \in J$.
Corollary to R2071: Isotonicity of set union. $\square$