Subset of Ordinals has Minimal Element

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Theorem

Let $A$ be an ordinal (we shall allow $A$ to be a proper class).

Let $B$ be a nonempty subset of $A$.

Then $B$ has an $\Epsilon$-minimal element.

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That is:

$\exists x \in B: B \cap x = \O$

Proof

We have that $\Epsilon$ creates a well-ordering on any ordinal.

Also, the initial segments of $x$ are sets.

Therefore from Proper Well-Ordering Determines Smallest Elements:

$B$ has an $\Epsilon$-minimal element.

Sources

• 1971: Gaisi Takeuti and Wilson M. Zaring: Introduction to Axiomatic Set Theory: $\S 7.5$