The inequality $b \geqq v + r - 1$ for a balanced incomplete block design was proved by Bose  under the assumption of resolvability. In this note the inequality is proved without that assumption, but with the weaker assumption that $v = nk$.
"An Inequality for Balanced Incomplete Block Designs." Ann. Math. Statist. 31 (2) 520 - 522, June, 1960. https://doi.org/10.1214/aoms/1177705921