reserve X,Y,Z,Z1,Z2,D for set,x,y for object;
reserve SFX,SFY,SFZ for set;
reserve F,G for Subset-Family of D;
reserve P for Subset of D;

theorem Th43:
  for X, x being set, R being Subset-Family of X st x in X holds x
  in Intersect R iff for Y being set st Y in R holds x in Y
proof
  let X, x be set, R be Subset-Family of X;
  assume
A1: x in X;
  hereby
    assume
A2: x in Intersect R;
    let Y be set;
    assume
A3: Y in R;
    then Intersect R = meet R by Def9;
    hence x in Y by A2,A3,Def1;
  end;
  assume
A4: for Y being set st Y in R holds x in Y;
  per cases;
  suppose
A5: R <> {};
    then x in meet R by A4,Def1;
    hence thesis by A5,Def9;
  end;
  suppose
    R = {};
    hence thesis by A1,Def9;
  end;
end;
