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_1:90;
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 ORDINAL1:4; :: 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