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 Th27:
  (ex x st Class(EqR,x) = X) implies EqR = nabla X
proof
  given x such that
A1: Class(EqR,x) = X;
  [:X,X:] c= EqR
  proof
    let y,z be object;
    assume
A2: [y,z] in [:X,X:];
    then z in Class(EqR,x) by A1,ZFMISC_1:87;
    then [z,x] in EqR by Th19;
    then
A3: [x,z] in EqR by Th6;
    y in Class(EqR,x) by A1,A2,ZFMISC_1:87;
    then [y,x] in EqR by Th19;
    hence thesis by A3,Th7;
  end;
  hence thesis;
end;
