theorem
  not X is finite iff ex Y st Y c= X & card Y = omega
proof
  thus not X is finite implies ex Y st Y c= X & card Y = omega
  proof
    assume not X is finite;
    then not card X in omega;
    then
A1: omega c= card X by CARD_1:4;
    card X,X are_equipotent by CARD_1:def 2;
    then consider f such that
A2: f is one-to-one and
A3: dom f = card X and
A4: rng f = X;
    take Y = f.:(omega);
    thus Y c= X by A4,RELAT_1:111;
    omega,Y are_equipotent
    proof
      take f|(omega);
      thus thesis by A1,A2,A3,FUNCT_1:52,RELAT_1:62,115;
    end;
    hence thesis by CARD_1:def 2;
  end;
  given Y such that
A5: Y c= X and
A6: card Y = omega;
  thus thesis by A5,A6;
end;
