let x1, x2, x3, x4, x5, x6, x7 be set ; :: thesis: card {x1,x2,x3,x4,x5,x6,x7} <= 7
( card {x1} = 1 & {x1,x2,x3,x4,x5,x6,x7} = {x1} \/ {x2,x3,x4,x5,x6,x7} & {x1} is finite & {x2,x3,x4,x5,x6,x7} is finite & card {x2,x3,x4,x5,x6,x7} <= 6 ) by Th73, CARD_1:50, ENUMSET1:56;
then ( 1 + (card {x2,x3,x4,x5,x6,x7}) <= 1 + 6 & card {x1,x2,x3,x4,x5,x6,x7} <= 1 + (card {x2,x3,x4,x5,x6,x7}) ) by Th62, XREAL_1:9;
hence card {x1,x2,x3,x4,x5,x6,x7} <= 7 by XXREAL_0:2; :: thesis: verum