reserve X, Y, Z for set, x, y, z for object;

theorem Th3:
  X meets Y iff ex x st x in X & x in Y
proof
  hereby
    assume X meets Y;
    then X /\ Y <> {};
    then X /\ Y is not empty by Lm1;
    then consider x such that
A1: x in X /\ Y;
    take x;
    thus x in X & x in Y by A1,Def4;
  end;
  given x such that
A2: x in X & x in Y;
  x in X /\ Y by A2,Def4;
  then X /\ Y <> {} by Def1;
  hence thesis;
end;
