reserve E,X,Y,x for set;
reserve A,B,C for Subset of E;

theorem Th23:
  A misses B iff A c= B`
proof
  thus A misses B implies A c= B`
  proof
    assume
A1: A misses B;
    let x be object;
    assume
A2: x in A;
    then
A3: not x in B by A1,XBOOLE_0:3;
    x in E by A2,Lm1;
    hence thesis by A3,XBOOLE_0:def 5;
  end;
  assume
A4: A c= B`;
  assume A meets B;
  then consider x being object such that
A5: x in A and
A6: x in B by XBOOLE_0:3;
  x in E \ B by A4,A5;
  hence thesis by A6,XBOOLE_0:def 5;
end;
