
theorem Th1:
  for A, B being set, x being object holds
    A = B \ {x} & x in B implies B \ A = {x}
proof
  let A, B be set;
  let x be object;
  assume that
  A1: A = B \ {x} and
  A2: x in B;
  reconsider A as Subset of B by A1;
  reconsider iks = {x} as Subset of B by A2, ZFMISC_1:31;
  A = iks` by A1, SUBSET_1:def 4;
  then A` = iks;
  hence thesis by SUBSET_1:def 4;
end;
