Set is Subset of Union/Family of Sets/Proof 2

Theorem

Let $\family {S_\alpha}_{\alpha \mathop \in I}$ be a family of sets indexed by $I$.

Then:

$\ds \forall \beta \in I: S_\beta \subseteq \bigcup_{\alpha \mathop \in I} S_\alpha$

where $\ds \bigcup_{\alpha \mathop \in I} S_\alpha$ is the union of $\family {S_\alpha}$.

Let $\beta \in I$ be arbitrary.

Then:

$\ds \beta$

$\in$

$\ds I$

$\ds \leadsto \ \ $

$\ds \set \beta$

$\subseteq$

$\ds I$

$\ds \leadsto \ \ $

$\ds \bigcup \set {S_\beta}$

$\subseteq$

$\ds \bigcup_{\alpha \mathop \in I} S_\alpha$

$\ds \leadsto \ \ $

$\ds S_\beta$

$\subseteq$

$\ds \bigcup_{\alpha \mathop \in I} S_\alpha$

Definition of Set Union

So it follows that:

$\ds \forall \beta \in I: S_\beta \subseteq \bigcup_{\alpha \mathop \in I} S_\alpha$

$\blacksquare$