Union of Blocks is Set of Points/Corollary

Corollary to Union of Blocks is Set of Points
Let $\struct {X, \BB}$ be a balanced incomplete block design.

Then the set union of all the subset elements in $\BB$ is precisely $X$.

Proof
By the definition of balanced incomplete block design, $\struct {X, \BB}$ is itself a type of pairwise balanced design.

Thus the corollary follows from Union of Blocks is Set of Points.