set MyOmega = {{},{1,2},{3,4},{1,2,3,4}};
{{},{1,2},{3,4},{1,2,3,4}} is Subset-Family of {1,2,3,4}
proof
for
x being
object st
x in {{},{1,2},{3,4},{1,2,3,4}} holds
x in bool {1,2,3,4}
proof
let x be
object ;
( x in {{},{1,2},{3,4},{1,2,3,4}} implies x in bool {1,2,3,4} )
assume
x in {{},{1,2},{3,4},{1,2,3,4}}
;
x in bool {1,2,3,4}
end;
hence
{{},{1,2},{3,4},{1,2,3,4}} is
Subset-Family of
{1,2,3,4}
by TARSKI:def 3;
verum
end;
then reconsider MyOmega = {{},{1,2},{3,4},{1,2,3,4}} as Subset-Family of {1,2,3,4} ;
A1:
MyOmega is compl-closed
MyOmega is sigma-multiplicative
by Lm8;
hence
{{},{1,2},{3,4},{1,2,3,4}} is SigmaField of {1,2,3,4}
by A1; verum