let X1, X2, X3 be set ; :: thesis: ( X1 <> {} & X2 <> {} & X3 <> {} implies for x being Element of [:X1,X2,X3:] holds
( x <> x `1 & x <> x `2 & x <> x `3 ) )
assume A1: ( X1 <> {} & X2 <> {} & X3 <> {} ) ; :: thesis: for x being Element of [:X1,X2,X3:] holds
( x <> x `1 & x <> x `2 & x <> x `3 )
let x be Element of [:X1,X2,X3:]; :: thesis: ( x <> x `1 & x <> x `2 & x <> x `3 )
set Y9 = {(x `1 ),(x `2 )};
set Y = {{(x `1 ),(x `2 )},{(x `1 )}};
set X9 = {{{(x `1 ),(x `2 )},{(x `1 )}},(x `3 )};
set X = {{{{(x `1 ),(x `2 )},{(x `1 )}},(x `3 )},{{{(x `1 ),(x `2 )},{(x `1 )}}}};
A2: x = [(x `1 ),(x `2 ),(x `3 )] by A1, Th48
.= [[(x `1 ),(x `2 )],(x `3 )] ;
then ( ( x = x `1 or x = x `2 ) implies ( {{{{(x `1 ),(x `2 )},{(x `1 )}},(x `3 )},{{{(x `1 ),(x `2 )},{(x `1 )}}}} in {(x `1 ),(x `2 )} & {(x `1 ),(x `2 )} in {{(x `1 ),(x `2 )},{(x `1 )}} & {{(x `1 ),(x `2 )},{(x `1 )}} in {{{(x `1 ),(x `2 )},{(x `1 )}},(x `3 )} & {{{(x `1 ),(x `2 )},{(x `1 )}},(x `3 )} in {{{{(x `1 ),(x `2 )},{(x `1 )}},(x `3 )},{{{(x `1 ),(x `2 )},{(x `1 )}}}} ) ) by TARSKI:def 2;
hence ( x <> x `1 & x <> x `2 ) by ORDINAL1:4; :: thesis: x <> x `3
x `3 = x `2 by A1, Th50;
hence x <> x `3 by A2, Th20; :: thesis: verum