Ordered pair

If an ordered pair were defined (in terms of sets) as (x,y):=a,a,b then the "first element" of an ordered pair S could be defined as CAR(S) where CAR(S) = x if and only if (∀y∈S.,x∈y). Likewise, the "second element" of S could be defined as CDR(S) where CDR(S) = x if and only if (∃y∈S.,(∃z∈y.,x∈z)). If the two elements happened to be equal, then the ordered pair would still have cardinality two as would be naturally expected.