
theorem Th19:
for X, x being set st card union X = 1 holds X is pairfree
proof
 let G, x be set;
 assume A1: card union G = 1;
 assume not G is pairfree;
 then PairsOf G <> {};
 then consider e being object such that
A2: e in PairsOf G by XBOOLE_0:def 1;
   consider x, y being set such that
A3: x <> y and
A4: x in union G and
A5: y in union G and
   e = {x, y} by A2,Th11;
   consider w being object such that
A6: union G = {w} by A1,CARD_2:42;
   x = w by A4,A6,TARSKI:def 1;
  hence contradiction by A3,A5,A6,TARSKI:def 1;
end;
