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
  for A,B,X being set holds ( X c= union (A \/ B) & for Y being set st Y
  in B holds Y misses X ) implies X c= union A
proof
  let A,B,X be set;
  assume X c= union (A \/ B);
  then X c= union (A \/ B) /\ X by XBOOLE_1:19;
  then X c= (union A \/ union B) /\ X by ZFMISC_1:78;
  then
A1: X c= (union A /\ X) \/ (union B /\ X) by XBOOLE_1:23;
  assume for Y st Y in B holds Y misses X;
  then union B misses X by ZFMISC_1:80;
  then
A2: union B /\ X = {};
  union A /\ X c= union A by XBOOLE_1:17;
  hence thesis by A2,A1;
end;
