reserve A,B,C,Y,x,y,z for set, U, D for non empty set,
X for non empty Subset of D, d,d1,d2 for Element of D;
reserve P,Q,R for Relation, g for Function, p,q for FinSequence;
reserve f for BinOp of D, i,m,n for Nat;
reserve X for set, f for Function;
reserve U1,U2 for non empty set;

theorem Th29:
(X \+\ Y={} iff X=Y) & (X\Y={} iff X c= Y) &
for x being object holds ({x}\Y={} iff x in Y)
proof
set Z=X \+\ Y, Z1=X\Y, Z2=Y\X;
thus Z={} implies X=Y
proof
assume Z={}; then Z1={} & Z2={}; then
X c= Y & Y c= X by XBOOLE_1:37; hence thesis;
end;
thus X=Y implies Z={};
thus X\Y={} iff X c= Y by XBOOLE_1:37;
thus thesis by ZFMISC_1:60;
end;
