let X1, X2, X3, X4 be set ; :: thesis: ( X1 <> {} & X2 <> {} & X3 <> {} & X4 <> {} implies for x being Element of [:X1,X2,X3,X4:] holds
( x <> x `1 & x <> x `2 & x <> x `3 & x <> x `4 ) )
assume A1: ( X1 <> {} & X2 <> {} & X3 <> {} & X4 <> {} ) ; :: thesis: for x being Element of [:X1,X2,X3,X4:] holds
( x <> x `1 & x <> x `2 & x <> x `3 & x <> x `4 )
then A2: [:X1,X2:] <> {} by ZFMISC_1:113;
let x be Element of [:X1,X2,X3,X4:]; :: thesis: ( x <> x `1 & x <> x `2 & x <> x `3 & x <> x `4 )
set Z' = {(x `1 ),(x `2 )};
set Z = {{(x `1 ),(x `2 )},{(x `1 )}};
set Y' = {{{(x `1 ),(x `2 )},{(x `1 )}},(x `3 )};
set Y = {{{{(x `1 ),(x `2 )},{(x `1 )}},(x `3 )},{{{(x `1 ),(x `2 )},{(x `1 )}}}};
set X' = {{{{{(x `1 ),(x `2 )},{(x `1 )}},(x `3 )},{{{(x `1 ),(x `2 )},{(x `1 )}}}},(x `4 )};
set X = {{{{{{(x `1 ),(x `2 )},{(x `1 )}},(x `3 )},{{{(x `1 ),(x `2 )},{(x `1 )}}}},(x `4 )},{{{{{(x `1 ),(x `2 )},{(x `1 )}},(x `3 )},{{{(x `1 ),(x `2 )},{(x `1 )}}}}}};
x = [(x `1 ),(x `2 ),(x `3 ),(x `4 )] by A1, Th60
.= [[{{(x `1 ),(x `2 )},{(x `1 )}},(x `3 )],(x `4 )] by TARSKI:def 5
.= [{{{{(x `1 ),(x `2 )},{(x `1 )}},(x `3 )},{{{(x `1 ),(x `2 )},{(x `1 )}}}},(x `4 )] by TARSKI:def 5
.= {{{{{{(x `1 ),(x `2 )},{(x `1 )}},(x `3 )},{{{(x `1 ),(x `2 )},{(x `1 )}}}},(x `4 )},{{{{{(x `1 ),(x `2 )},{(x `1 )}},(x `3 )},{{{(x `1 ),(x `2 )},{(x `1 )}}}}}} by TARSKI:def 5 ;
then ( ( x = x `1 or x = x `2 ) implies ( {{{{{{(x `1 ),(x `2 )},{(x `1 )}},(x `3 )},{{{(x `1 ),(x `2 )},{(x `1 )}}}},(x `4 )},{{{{{(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 )}}}} & {{{{(x `1 ),(x `2 )},{(x `1 )}},(x `3 )},{{{(x `1 ),(x `2 )},{(x `1 )}}}} in {{{{{(x `1 ),(x `2 )},{(x `1 )}},(x `3 )},{{{(x `1 ),(x `2 )},{(x `1 )}}}},(x `4 )} & {{{{{(x `1 ),(x `2 )},{(x `1 )}},(x `3 )},{{{(x `1 ),(x `2 )},{(x `1 )}}}},(x `4 )} in {{{{{{(x `1 ),(x `2 )},{(x `1 )}},(x `3 )},{{{(x `1 ),(x `2 )},{(x `1 )}}}},(x `4 )},{{{{{(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 ORDINAL1:6; :: thesis: ( x <> x `3 & x <> x `4 )
reconsider x' = x as Element of [:[:X1,X2:],X3,X4:] by Th54;
A3: x `3 = (x `1 ) `2 by A1, Th61
.= x' `2 by A1, A2, Th50 ;
x `4 = x `2 by A1, Th61
.= x' `3 by A1, A2, Th50 ;
hence ( x <> x `3 & x <> x `4 ) by A1, A2, A3, Th51; :: thesis: verum