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 ORDINAL1:2; :: thesis: x <> x `3
x `3 = x `2 by A1, Th50;
hence x <> x `3 by A2, Th20; :: thesis: verum