reserve X,Y,Z for set, x,y,z for object;
reserve i,j for Nat;
reserve A,B,C for Subset of X;
reserve R,R1,R2 for Relation of X;
reserve AX for Subset of [:X,X:];
reserve SFXX for Subset-Family of [:X,X:];
reserve EqR,EqR1,EqR2,EqR3 for Equivalence_Relation of X;

theorem Th31:
  X = {} implies Class EqR = {}
proof
  assume that
A1: X = {} and
A2: Class EqR <> {};
  set z = the Element of Class EqR;
  z is Subset of X by A2,TARSKI:def 3;
  then ex x st x in X & z = Class(EqR,x) by A2,Def3;
  hence contradiction by A1;
end;
