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

Result R1035 on D80: Power set

Power set is closed under intersections

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 \implies \quad \bigcap_{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 \bigcap_{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 $j \in J$.

Direct corollary to R2079: Isotonicity of set intersection. $\square$