DeMorgan's laws for arbitrary unions and intersections

Fix a set \(X\) and a family \(\{A_i\}_{i\in I}\) of subsets of \(X\). The arbitrary DeMorgan laws are \[ X-\bigcup_{i\in I}A_i = \bigcap_{i\in I}(X-A_i) \] and \[ X-\bigcap_{i\in I}A_i = \bigcup_{i\in I}(X-A_i). \]

For the first identity, \[ \begin{aligned} x\in X-\bigcup_iA_i &\iff x\in X\text{ and }x\notin A_i\text{ for every }i\\ &\iff x\in X-A_i\text{ for every }i\\ &\iff x\in\bigcap_i(X-A_i). \end{aligned} \] For the second, \[ \begin{aligned} x\in X-\bigcap_iA_i &\iff x\in X\text{ and it is not true that }x\in A_i\text{ for every }i\\ &\iff x\in X\text{ and }x\notin A_i\text{ for at least one }i\\ &\iff x\in\bigcup_i(X-A_i). \end{aligned} \]