On the Order on a Special Polygon

One can prove the following propositions: (1) For all sets A, B, C, p such that A ∩ B ⊆ {p} and p ∈ C and C misses B holds A ∪ C misses B. (2) For all sets A, B, C, p such that A ∩ C = {p} and p ∈ B and B ⊆ C holds A ∩ B = {p}. (3) For all sets A, B such that for every set y such that y ∈ B holds A misses y holds A misses ⋃ B. (4) For all sets A, B such that for all sets x, y such that x ∈ A and y ∈ B holds x misses y holds ⋃ A misses ⋃ B.