# Does proving $A ∪ (B ∩ C) ⊆ (A ∪ B) ∩ (A ∪ C)$ prove the distributive law of sets?

Does proving $A ∪ (B ∩ C) ⊆ (A ∪ B) ∩ (A ∪ C)$ prove the distributive law of sets?

I know the proof for A ∪ (B ∩ C) ⊆ (A ∪ B) ∩ (A ∪ C) but not the distributive property of sets but I believe it satisfies the distributive law? Is it or is it not?

• What is the distributive law? – Jacob Wakem Apr 24 '17 at 15:24
• @Alephnull $A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)$ – mathewconway Apr 24 '17 at 15:26
• Actually that inequality you are referring to is valid in any lattice, and in particular is valid for sets. It is the reverse inclusion that proves the distributive law. – amrsa Apr 24 '17 at 15:48