ThmDex – An index of mathematical definitions, results, and conjectures.

Result R4166 on D77: Set union

Superset of finite union iff superset of every set in the union

Formulation 0

Let $X, E_1, \ldots, E_N$ each be a D11: Set such that

(i)	$\bigcup_{n = 1}^N E_n$ is the D77: Set union of $E_1, \ldots, E_N$

Then \begin{equation} \bigcup_{n = 1}^N E_n \subseteq X \quad \iff \quad E_1 \subseteq X, \; \ldots, \; E_N \subseteq X \end{equation}

Proofs

Proof 0

Let $X, E_1, \ldots, E_N$ each be a D11: Set such that

(i)	$\bigcup_{n = 1}^N E_n$ is the D77: Set union of $E_1, \ldots, E_N$