In set theory, how does one prove the principle of absorption?

One approach is by extensionality and the definition of subset.Suppose z∈X∩(X∪Y)z∈X∩(X∪Y) . Then z∈Xz∈X , so X∩(X∪Y)⊆XX∩(X∪Y)⊆X .Now, suppose z∈Xz∈X . It follows that z∈X∪Yz∈X∪Y , and therefore also z∈X∩(X∪Y)z∈X∩(X∪Y) . Hence X⊆X∩(X∪Y)X⊆X∩(X∪Y) , and by extensionality, X∩(X∪Y)=X. □