It turns out that the occurence of forbidden self-contained sets is only a technical problem and we can circumvent it by skillfully defining a contained relation, as follows:
Let $X$ be a set. The contained relation $\in_X\subseteq X \times X$ is a relation defined by $$\in_X:=\{(a,b)\in X\times X\mid a\in b\}.$$
Corollaries: 1
Definitions: 2 3
Examples: 4
Motivations: 5
Proofs: 6 7 8 9
Propositions: 10 11