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;
reserve X for non empty set,
  x for Element of X;

theorem Th39:
  for X being non empty set holds {X} is a_partition of X
proof
  let X be non empty set;
  reconsider A1 = {X} as Subset-Family of X by ZFMISC_1:68;
A1: for A being Subset of X st A in A1 holds A <> {} & for B being Subset of
  X st B in A1 holds A = B or A misses B
  proof
    let A be Subset of X;
    assume
A2: A in A1;
    hence A <> {} by TARSKI:def 1;
    let B be Subset of X;
    assume B in A1;
    then B = X by TARSKI:def 1;
    hence thesis by A2,TARSKI:def 1;
  end;
  union A1 = X;
  hence thesis by A1,Def4;
end;
