now

let x be set ; :: thesis: ( ( not x in {{},<*0*>,<*1*>} or x in { <*n*> where n is Element of NAT : n < 2 } or x in {{}} ) & ( ( x in { <*n*> where n is Element of NAT : n < 2 } or x in {{}} ) implies x in {{},<*0*>,<*1*>} ) )
thus ( not x in {{},<*0*>,<*1*>} or x in { <*n*> where n is Element of NAT : n < 2 } or x in {{}} ) :: thesis: ( ( x in { <*n*> where n is Element of NAT : n < 2 } or x in {{}} ) implies x in {{},<*0*>,<*1*>} )

proof

assume x in {{},<*0*>,<*1*>} ; :: thesis: ( x in { <*n*> where n is Element of NAT : n < 2 } or x in {{}} )
then ( x = {} or x = <*0*> or x = <*1*> ) by ENUMSET1:def 1;
hence ( x in { <*n*> where n is Element of NAT : n < 2 } or x in {{}} ) by TARSKI:def 1; :: thesis: verum

end;

assume A4: ( x in { <*n*> where n is Element of NAT : n < 2 } or x in {{}} ) ; :: thesis: x in {{},<*0*>,<*1*>}

now

per cases ( x in { <*n*> where n is Element of NAT : n < 2 } or x in {{}} ) by A4;

suppose x in { <*n*> where n is Element of NAT : n < 2 } ; :: thesis: x in {{},<*0*>,<*1*>}

then consider n being Element of NAT such that
A7: x = <*n*> and
A8: n < 2 ;
( n = 0 or n = 1 ) by A8, NAT_1:27;
hence x in {{},<*0*>,<*1*>} by A7, ENUMSET1:def 1; :: thesis: verum

end;

suppose x in {{}} ; :: thesis: x in {{},<*0*>,<*1*>}

then x = {} by TARSKI:def 1;
hence x in {{},<*0*>,<*1*>} by ENUMSET1:def 1; :: thesis: verum

end;

end;

end;

hence x in {{},<*0*>,<*1*>} ; :: thesis: verum

end;

hence elementary_tree 2 = {{},<*0*>,<*1*>} by XBOOLE_0:def 3; :: thesis: verum