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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?

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  • $\begingroup$ What is the distributive law? $\endgroup$ – Jacob Wakem Apr 24 '17 at 15:24
  • $\begingroup$ @Alephnull $A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)$ $\endgroup$ – mathewconway Apr 24 '17 at 15:26
  • $\begingroup$ 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. $\endgroup$ – amrsa Apr 24 '17 at 15:48
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You are halfway to proving the distributive law. You also need to show the inclusion the other way.

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