reserve i,j,n,k,m for Nat,
     a,b,x,y,z for object,
     F,G for FinSequence-yielding FinSequence,
     f,g,p,q for FinSequence,
     X,Y for set,
     D for non empty set;

theorem Th17:
  Y in Ext(X,x,y) & not y in union X implies (x in Y iff y in Y)
proof
  assume that
A1: Y in Ext(X,x,y) and
A2:  not y in union X;
  per cases by A1,XBOOLE_0:def 3;
  suppose Y in {A\/{y} where A is Element of X: x in A};
    then ex A be Element of X st Y =A\/{y} & x in A;
    hence thesis by ZFMISC_1:136;
  end;
  suppose Y in {A where A is Element of X : not x in A & A in X};
    then ex A be Element of X st
    Y=A & not x in A & A in X;
    hence thesis by A2,TARSKI:def 4;
  end;
end;
