set Y = { x where x is Element of X : x is atom } ;
for z being Element of X st z \ (0. X) = 0. X holds
z = 0. X by Th2;
then 0. X is atom by Def14;
then A1: 0. X in { x where x is Element of X : x is atom } ;

now

let y be set ; :: thesis: ( y in { x where x is Element of X : x is atom } implies y in the carrier of X )
assume y in { x where x is Element of X : x is atom } ; :: thesis: y in the carrier of X
then consider x being Element of X such that
A2: ( y = x & x is atom ) ;
thus y in the carrier of X by A2; :: thesis: verum

end;

hence { x where x is Element of X : x is atom } is non empty Subset of X by A1, TARSKI:def 3; :: thesis: verum