reserve u,v,x,x1,x2,y,y1,y2,z,p,a for object,
        A,B,X,X1,X2,X3,X4,Y,Y1,Y2,Z,N,M for set;
reserve e for object, X,X1,X2,Y1,Y2 for set;

theorem
  for X being set holds X is non trivial iff
   for x holds X\{x} is non empty
proof
  let X be set;
  hereby
    assume
A1: X is non trivial;
    let x be object;
    X <> {x} by A1;
    then consider y being object such that
A2: y in X and
A3: x <> y by A1,Lm12;
    not y in {x} by A3,TARSKI:def 1;
    hence X\{x} is non empty by A2,XBOOLE_0:def 5;
  end;
  assume
A4: for x holds X\{x} is non empty;
   X\{{}} c= X by XBOOLE_1:36;
   then X is non empty by A4;
   then consider x being object such that
A5:  x in X;
   X\{x} is non empty by A4;
   then consider y being object such that
A6:  y in X\{x};
   reconsider x,y as set by TARSKI:1;
  take x,y;
  thus x in X by A5;
   X\{x} c= X by XBOOLE_1:36;
  hence y in X by A6;
   x in {x} by TARSKI:def 1;
  hence x <> y by A6,XBOOLE_0:def 5;
end;
