Strict Ordering of Ordinals is Equivalent to Membership Relation

Theorem
Let $\On$ denote the class of all ordinals.

Let $<$ denote the (strict) usual ordering of $\On$.

Then:
 * $\forall \alpha, \beta \in \On: \alpha < \beta \iff \alpha \in \beta$

Necessary Condition
Let $\alpha \in \beta$.

Then from Ordinal is Transitive:
 * $\alpha \subseteq \beta$

But if $\alpha = \beta$ we would have $\alpha \in \alpha$.

This is contrary to Ordinal is not Element of Itself.

Hence we have:
 * $\alpha \subseteq \beta$

and:
 * $\alpha \ne \beta$

That is:
 * $\alpha \subsetneqq \beta$

Hence by definition of the (strict) usual ordering of $\On$:
 * $\alpha < \beta$

Sufficient Condition
Let $\alpha < \beta$.

Then by Sandwich Principle for $g$-Towers: Corollary 1:
 * $\alpha^+ \subseteq \beta$

where $\alpha^+$ denotes the successor set of $\alpha$.

By definition of successor set:
 * $\alpha^+ = \alpha \cup \set \alpha$

and so:
 * $\alpha \in \alpha^+$

That is:
 * $\alpha \in \beta$