let x1, x2, x3, x4, x5 be set ; :: thesis: card {x1,x2,x3,x4,x5} <= 5
( card {x1} = 1 & {x1,x2,x3,x4,x5} = {x1} \/ {x2,x3,x4,x5} & {x1} is finite & {x2,x3,x4,x5} is finite & card {x2,x3,x4,x5} <= 4 ) by Th71, CARD_1:50, ENUMSET1:47;
then ( 1 + (card {x2,x3,x4,x5}) <= 1 + 4 & card {x1,x2,x3,x4,x5} <= 1 + (card {x2,x3,x4,x5}) ) by Th62, XREAL_1:9;
hence card {x1,x2,x3,x4,x5} <= 5 by XXREAL_0:2; :: thesis: verum