defpred S₁[ set ] means $1 is non empty IntervalSet of U;
ex X being set st
for x being set holds
( x in X iff ( x in bool (bool U) & S₁[x] ) ) from XBOOLE_0:sch 1();
then consider X being set such that
A1: for x being set holds
( x in X iff ( x in bool (bool U) & S₁[x] ) ) ;
consider x being Element of U;
reconsider E = {x} as Subset of U by ZFMISC_1:37;
{E} is non empty IntervalSet of U by ThB1;
then reconsider X = X as non empty set by A1;
take X ; :: thesis: for x being set holds
( x in X iff x is non empty IntervalSet of U )
thus for x being set holds
( x in X iff x is non empty IntervalSet of U ) by A1; :: thesis: verum