Catalysis of the reaction: beta-D-fructopyranosyl-(2->6)-[D-fructofuranosyl-(2->6)]n-D-fructofuranoside = beta-D-fructopyranosyl-(2->6)-[D-fructofuranosyl-(2->6)](n-1)-D-fructofuranoside + di-beta-D-fructofuranose 2,6':2',6-dianhydride. This reaction is the production of di-beta-D-fructofuranose 2,6':2',6-dianhydride (DFA IV) by successively eliminating the diminishing (2->6)-beta-D-fructan (levan) chain from the terminal D-fructosyl-D-fructosyl disaccharide.