Class of sets closed under complements properties

Theorem statement

If [ilmath]\mathcal{A} [/ilmath] is a system of subsets of [ilmath]\Omega[/ilmath] such that^[1]:

[math]\forall A\in\mathcal{A}[A^c\in\mathcal{A}][/math] where denotes the complement of [ilmath]A[/ilmath] - That is to say "[ilmath]\mathcal{A} [/ilmath] is closed under complements"

Then we have:^{[Note 1]}

[ilmath]\mathcal{A} [/ilmath] is [ilmath]\cap[/ilmath]-closed [ilmath]\iff[/ilmath] [ilmath]\mathcal{A} [/ilmath] is [ilmath]\cup[/ilmath]-closed
[ilmath]\mathcal{A} [/ilmath] is [ilmath]\sigma[/ilmath]-[ilmath]\cap[/ilmath]-closed [ilmath]\iff[/ilmath] [ilmath]\mathcal{A} [/ilmath] is [ilmath]\sigma[/ilmath]-[ilmath]\cup[/ilmath]-closed

Proof:

TODO: see ^[1] page 2 if help needed - basically is just De Morgan's laws