
theorem Th13:
for X being set, x, y being object
 st {x, y} in PairsOf X holds x <> y & x in union X & y in union X
proof
  let G be set, a, b be object;
  assume {a, b} in PairsOf G;
   then consider x, y being set such that
  A1: x <> y and
  A2: x in union G & y in union G and
  A3: {a,b} = {x, y} by Th11;
    a = x & b = y or a = y & b = x by A3,ZFMISC_1:6;
  hence thesis by A2,A1;
end;
