A combinatorial proof of a more general inequality is [given][1] by Douglas Woodall.
Woodall, Douglas R., A note on Fisher's inequality, J. Comb. Theory, Ser. A 77, No.1, 171-176, TA962729 (1997). ZBL0878.05011. [1]: https://pdfs.semanticscholar.org/f8ac/82fa89f026a5ced840f97086f4b40d370bbd.pdf