Transitive Set is Proper Subset of Ordinal iff Element of Ordinal
Theorem
Let $A$ be an ordinal.
Let $B$ be a transitive set.
Then:
- $B \subsetneq A \iff B \in A$
Corollary
Let $A$ and $B$ be ordinals.
Then:
- $A \subsetneq B \iff A \in B$
Proof
Necessary Condition
Suppose that $B \in A$.
From Ordinal is Transitive, it follows that $B \subseteq A$.
Also, $B \ne A$ by Ordinal is not Element of Itself.
Therefore, $B \subsetneq A$, as desired.
$\Box$
Sufficient Condition
Suppose that $B \subsetneq A$.
By the definition of set equality, the set difference $A \setminus B$ is non-empty.
By the definition of a strict well-ordering, there exists a minimal element $x$ of $A \setminus B$.
As $x \in A$, it follows from Ordinal is Transitive that $x \subseteq A$.
Next, since the strict well-ordering on $A$ is given by the epsilon restriction $\Epsilon \! \restriction_A$, it follows by the definition of a minimal element that:
- $\forall y \in A \setminus B: y \notin x$
Therefore:
| \(\ds \O\) | \(=\) | \(\ds \paren {A \setminus B} \cap x\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \paren {A \cap x} \setminus B\) | Intersection with Set Difference is Set Difference with Intersection | |||||||||||
| \(\ds \) | \(=\) | \(\ds x \setminus B\) | Intersection with Subset is Subset | |||||||||||
| \(\ds \leadsto \ \ \) | \(\ds x\) | \(\subseteq\) | \(\ds B\) | by Set Difference with Superset is Empty Set |
Suppose that $z \in B$.
Since $B \subset A$, it follows that $z \in A$.
Recall that the strict well-ordering on $A$ is given by the epsilon restriction $\Epsilon \! \restriction_A$.
From the Trichotomy Law (Ordering), it follows that $z \in x$ or $x = z$ or $x \in z$.
If $x = z$ or $x \in z$, then it follows by the transitivity of $B$ that $x \in B$.
This contradicts the definition of $x$.
Hence, $z \in x$.
That is, $B \subseteq x$.
We have shown that $x \subseteq B$ and $B \subseteq x$.
By definition of set equality:
- $B = x \in A$
$\blacksquare$
Also see
Sources
- 1971: Gaisi Takeuti and Wilson M. Zaring: Introduction to Axiomatic Set Theory: $\S 7.7$