defpred S₁[ object ] means ex E being finite Subset of I st $1 in sigUn (F,E);
consider X being set such that
A1: for x being object holds
( x in X iff ( x in bool Omega & S₁[x] ) ) from XBOOLE_0:sch 1();
for x being object st x in X holds
x in bool Omega by A1;
then reconsider X = X as Subset-Family of Omega by TARSKI:def 3;
take X ; :: thesis: for S being Subset of Omega holds
( S in X iff ex E being finite Subset of I st S in sigUn (F,E) )
let S be Subset of Omega; :: thesis: ( S in X iff ex E being finite Subset of I st S in sigUn (F,E) )
thus ( S in X iff ex E being finite Subset of I st S in sigUn (F,E) ) by A1; :: thesis: verum