As reference for others who like me may have been unclear about the term "vacuously true" used in the thread:

(If A then B) is vacuously true if hypothesis A is false.

For example: Let E be the empty set and S any set. Then E is a subset of S:

(If x ϵ E then x ϵ S) is vacuously true because x ϵ E is false. (Reference for others who may also have been un)

Note the last 2 lines in the truth table from post #26 are vacuously true:

Found a clear, precise, non-trivial definition of anti-symmetry in Kelly (General Topology), pg 9: “R is anti-symmetric iff it is never the case that both xRy and yRx. “

* parentheses added to clarify. Pointed out by Deveno.