I would rather remove the assumption that the property of 'being a member of itself' must apply to the whole set if it applies to each member of the set. The set of all sets with only one member has more than one member.
A compound statement may be split into one statement which is false and another which
is true. The liar paradox can be split into 'the following statement is false' and 'the previous statement is false', and when one of those two statements is false then the other statement is true, without any contradiction.
'the following statement is false' and 'the previous statement is true' may be split into '1)statements 3 and 4 are both true' '2) statement 4 is false', '3)statement 1 is true' and '4) statements 1 and 2 are both true', which can be non-contradictory when 1,3,4 are false and 2 is true.
If what I am typing is contradictory then I have made a mistake.
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