defpred S₁[ set ] means ex E being finite Subset of I st $1 = sigUn (F,(I \ E));
consider X being set such that
A1: for x being set holds
( x in X iff ( x in bool (bool Omega) & S₁[x] ) ) from XFAMILY:sch 1();
A2: not X is empty

proof

set Ie = I \ ({} I);
sigUn (F,(I \ ({} I))) in X by A1;
hence not X is empty ; :: thesis: verum

end;

for x being object st x in X holds
x in bool (bool Omega) by A1;
then reconsider X = X as non empty Subset-Family of (bool Omega) by A2, TARSKI:def 3;
take X ; :: thesis: for S being Subset-Family of Omega holds
( S in X iff ex E being finite Subset of I st S = sigUn (F,(I \ E)) )
let S be Subset-Family of Omega; :: thesis: ( S in X iff ex E being finite Subset of I st S = sigUn (F,(I \ E)) )
thus ( S in X iff ex E being finite Subset of I st S = sigUn (F,(I \ E)) ) by A1; :: thesis: verum