
theorem
  for X being set st X is c=filtered for a,b being set st a in X & b in
  X ex c being set st c c= a /\ b & c in X
proof
  let X be set;
  assume
A1: for Y being finite Subset of X ex a being set st (for y being set st
  y in Y holds a c= y) & a in X;
  let a,b be set;
  assume a in X & b in X;
  then {a,b} c= X by ZFMISC_1:32;
  then consider c being set such that
A2: for y being set st y in {a,b} holds c c= y and
A3: c in X by A1;
  take c;
  b in {a,b} by TARSKI:def 2;
  then
A4: c c= b by A2;
  a in {a,b} by TARSKI:def 2;
  then c c= a by A2;
  hence thesis by A3,A4,XBOOLE_1:19;
end;
