Trichotomy and well-ordering of the ordinals

At least one alternative holds: by [L1] either $\alpha \subseteq \beta$ or $\beta \subseteq \alpha$, and in the first case [L1] gives $\alpha \in \beta$ or $\alpha = \beta$, in the second $\beta \in \alpha$ or $\beta = \alpha$.

At most one alternative holds: $\alpha \in \beta$ together with $\alpha = \beta$ gives $\alpha \in \alpha$; $\beta \in \alpha$ together with $\alpha = \beta$ gives $\alpha \in \alpha$; and $\alpha \in \beta$ together with $\beta \in \alpha$ gives $\beta \subseteq \alpha$ by transitivity of $\alpha$ and hence $\alpha \in \alpha$; each contradicts [L1].

Least elements: fix $\alpha \in A$; if $\alpha \cap A = \emptyset$ then $\alpha$ is $\in$-least in $A$, because $\beta \in A$ with $\beta \in \alpha$ would lie in $\alpha \cap A$, so trichotomy leaves $\alpha \in \beta$ or $\alpha = \beta$; otherwise $\alpha \cap A$ is a nonempty subset of $\alpha$ and has an $\in$-least element $\gamma$ there, and $\gamma$ is $\in$-least in $A$, because $\beta \in A$ with $\beta \in \gamma$ would satisfy $\beta \in \alpha$ by transitivity of $\alpha$ and so lie in $\alpha \cap A$ strictly below $\gamma$, and trichotomy again leaves $\gamma \in \beta$ or $\gamma = \beta$.

On $A$ the relation $\alpha \le \beta :\iff \alpha \subseteq \beta$ satisfies the three axioms of [L2], since inclusion is reflexive, antisymmetric by extensionality, and transitive, so it is a partial order; it is total by [L1] and step 1.1; its strict part in the sense of [L2] is membership, since $\alpha \subseteq \beta$ with $\alpha \ne \beta$ is $\alpha \in \beta$ by [L1]; and every nonempty subset of $A$ has a least element by step 3.1, so $(A, \subseteq)$ is a well-order in the sense of [L3] and [L4] applies to it.

Exactly one of the three alternatives holds, every nonempty set of ordinals has an $\in$-least element, and every set of ordinals is well ordered by inclusion.