
theorem Th11:
for X being set, e being set
 st e in PairsOf X
  ex x, y being set st x <> y & x in union X & y in union X & e = {x, y}
proof
 let G be set, e be set; assume
A1: e in PairsOf G; then
   card e = 2 by Def1; then
   consider x, y being object such that
A2: x <> y and
A3: e = {x,y} by CARD_2:60;
   x in e & y in e by A3,TARSKI:def 2;
  then x in union G & y in union G by A1,TARSKI:def 4;
  hence thesis by A2,A3;
end;
