let X1, X2, X3, Y1, Y2, Y3 be set ; :: thesis: ( [:X1,X2,X3:] <> {} & [:X1,X2,X3:] = [:Y1,Y2,Y3:] implies ( X1 = Y1 & X2 = Y2 & X3 = Y3 ) )
assume A1: [:X1,X2,X3:] <> {} ; :: thesis: ( not [:X1,X2,X3:] = [:Y1,Y2,Y3:] or ( X1 = Y1 & X2 = Y2 & X3 = Y3 ) )
then A2: X3 <> {} by Th21;
( X1 <> {} & X2 <> {} ) by A1, Th21;
hence ( not [:X1,X2,X3:] = [:Y1,Y2,Y3:] or ( X1 = Y1 & X2 = Y2 & X3 = Y3 ) ) by A2, Th22; :: thesis: verum