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
  for X being non empty set holds
    SmallestPartition X = the set of all {x} where x is Element of X
proof
  let X be non empty set;
  set Y = the set of all {x} where x is Element of X;
  hereby
    let x be object;
    assume x in SmallestPartition X;
    then consider y being object such that
A1: y in X and
A2: x = Class(id X, y) by Def3;
    x = {y} by A1,A2,Th25;
    hence x in Y by A1;
  end;
  let x be object;
  assume x in Y;
  then consider y being Element of X such that
A3: x = {y};
  Class(id X, y) = x by A3,Th25;
  hence thesis by Def3;
end;
