
theorem Th51: :: AdjacentV00
  for G being _Graph, u,v being Vertex of G holds u in G
  .AdjacentSet({v}) iff u <> v & v,u are_adjacent
proof
  let G be _Graph, u,v be Vertex of G;
A1: v in {v} by TARSKI:def 1;
  hereby
    assume
A2: u in G.AdjacentSet({v});
    then consider x being Vertex of G such that
A3: x in {v} and
A4: u,x are_adjacent by Th49;
    x = v by A3,TARSKI:def 1;
    hence u <> v & v,u are_adjacent by A2,A3,A4,Th49;
  end;
  assume that
A5: u <> v and
A6: v,u are_adjacent;
  not u in {v} by A5,TARSKI:def 1;
  hence thesis by A6,A1;
end;
