why are statement 1 and statement 2 analogous for disjoint sets?
Statement 1 (Intersection): if $X \perp Y \mid Z$ and $X \perp Z \mid Y$ then $X \perp (Y,Z)$.
Statement 2: if $A \perp B \mid (C \cup D)$ and $A \perp C \mid (B \cup D)$ then $A \perp (B \cup C) \mid D$
Statement 2 is used as condition for Pearl and Paz theorem of equivalence of Markov properties for undirected graphs.