reserve A,B for Ordinal,
  K,M,N for Cardinal,
  x,x1,x2,y,y1,y2,z,u for object,X,Y,Z,X1,X2, Y1,Y2 for set,
  f,g for Function;

theorem
  x in X & X,Y are_equipotent implies Y <> {} &
   ex x st x in Y
proof
  assume
A1: x in X;
  given f such that
  f is one-to-one and
A2: dom f = X & rng f = Y;
  f.x in Y by A1,A2,FUNCT_1:def 3;
  hence Y <> {};
  take f.x;
  thus thesis by A1,A2,FUNCT_1:def 3;
end;
