reserve W,X,Y,Z for set,
  f,g for Function,
  a,x,y,z for set;
reserve u,v for Element of Tarski-Class(X),
  A,B,C for Ordinal,
  L for Sequence;

theorem Th26:
  (x in Tarski-Class X implies {x} in Tarski-Class X) &
  (x in Tarski-Class X & y in Tarski-Class X implies
    {x,y} in Tarski-Class X)
proof
  thus
Z1: now assume x in Tarski-Class X; then
    bool x in Tarski-Class X by Th4;
    hence {x} in Tarski-Class X by Th3,ZFMISC_1:68;
  end;
  assume that
A1: x in Tarski-Class X and
A2: y in Tarski-Class X;
A3: {x} in Tarski-Class X by Z1,A1;
 bool {x} = {{},{x}} by ZFMISC_1:24;
then A4: not {{},{x}},Tarski-Class X are_equipotent by A3,Th4,Th25;
  now
    assume
A5: x <> y;
    {{},{x}},{x,y} are_equipotent
    proof
      defpred C[object] means $1 = {};
      deffunc f(object) = x;
      deffunc g(object) = y;
      consider f such that
A6:  dom f = {{},{x}} & for z being object st z in {{},{x}} holds
      (C[z] implies f.z = f(z)) & (not C[z] implies f.z = g(z))
      from PARTFUN1:sch 1;
      take f;
      thus f is one-to-one
      proof
        let x1,x2 be object;
        assume that
A7:    x1 in dom f and
A8:    x2 in dom f;
A9:    x2 = {} or x2 = {x} by A6,A8,TARSKI:def 2;
A10:    x1 = {} implies f.x1 = x by A6,A7;
        x1 <> {} implies f.x1 = y by A6,A7;
        hence thesis by A5,A6,A7,A8,A9,A10,TARSKI:def 2;
      end;
      thus dom f = {{},{x}} by A6;
      thus rng f c= {x,y}
      proof
        let z be object;
        assume z in rng f; then
        ex u being object st u in dom f & z = f.u by FUNCT_1:def 3; then
        z = x or z = y by A6;
        hence thesis by TARSKI:def 2;
      end;
      let z be object;
      assume z in {x,y}; then
A11:  z = x or z = y by TARSKI:def 2;
A12:  {} in dom f by A6,TARSKI:def 2;
A13:  {x} in dom f by A6,TARSKI:def 2;
A14:  f.{} = x by A6,A12;
      f.{x} = y by A6,A13;
      hence thesis by A11,A12,A13,A14,FUNCT_1:def 3;
    end;
then A15: not {x,y},Tarski-Class X are_equipotent by A4,WELLORD2:15;
 {x,y} c= Tarski-Class X by A1,A2,ZFMISC_1:32;
    hence thesis by A15,Th5;
  end;
  hence thesis by A3,ENUMSET1:29;
end;
