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 that
A1: ( X1 <> {} & X2 <> {} ) and
A2: ( 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 )
let x be Element of [:X1,X2,X3,X4:]; :: thesis: ( x <> x `1 & x <> x `2 & x <> x `3 & x <> x `4 )
reconsider x9 = x as Element of [:[:X1,X2:],X3,X4:] by Th54;
A3: [:X1,X2:] <> {} by A1, ZFMISC_1:113;
A4: x `4 = x `2 by A1, A2, Th61
.= x9 `3 by A2, A3, Th50 ;
set Z9 = {(x `1 ),(x `2 )};
set Z = {{(x `1 ),(x `2 )},{(x `1 )}};
set Y9 = {{{(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 X9 = {{{{{(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, A2, Th60
.= {{{{{{(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 )}}}}}} ;
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 )
x `3 = (x `1 ) `2 by A1, A2, Th61
.= x9 `2 by A2, A3, Th50 ;
hence ( x <> x `3 & x <> x `4 ) by A2, A3, A4, Th51; :: thesis: verum