Halmos Naive Set Theory - Set Inclusion Antisymmetric
"Antisymmetric" doesn't mean what you think. A relation $R$ is antisymmetric if $aRb$ and $bRa$ implies that $a = b$. (If $aRb$ implies that $bRa$ cannot hold, the relation is called "irreflexive", which, confusingly is much stronger than "not reflexive".)