let x1, x2, x3, x4, x5, x6, x7, x8 be set ; :: thesis: card {x1,x2,x3,x4,x5,x6,x7,x8} <= 8
( {x1,x2,x3,x4,x5,x6,x7,x8} = {x1} \/ {x2,x3,x4,x5,x6,x7,x8} & card {x1} = 1 & {x1} is finite & {x2,x3,x4,x5,x6,x7,x8} is finite & card {x2,x3,x4,x5,x6,x7,x8} <= 7 ) by Th74, CARD_1:50, ENUMSET1:62;
then ( 1 + (card {x2,x3,x4,x5,x6,x7,x8}) <= 1 + 7 & card {x1,x2,x3,x4,x5,x6,x7,x8} <= 1 + (card {x2,x3,x4,x5,x6,x7,x8}) ) by Th62, XREAL_1:9;
hence card {x1,x2,x3,x4,x5,x6,x7,x8} <= 8 by XXREAL_0:2; :: thesis: verum