Consistently there is no non trivial ccc forcing notion with the Sacks or Laver property

Abstract

Boban Velickovic asked the following question: Is there a nontrivial forcing notion with the Sacks property which is also ccc? A ``definable'' variant of this question has been answered in [Sh:480] (math.LO/9303208): Every nontrivial Souslin forcing notion which has the Sacks property has an uncountable antichain. Here we show that it is consistent that every nontrivial forcing notion which has the Sacks property has an uncountable antichain. Independently, Velickovic has also proved the consistency of this statement.