Minimally Inductive Set is Minimal
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Theorem
The minimally inductive set $\omega$ is a subset of every inductive set.
Proof
Let $A$ be an inductive set.
Let $B$ be another arbitrary inductive set.
Then from Intersection is Subset, $A \cap B \subseteq A$.
From Intersection of Inductive Sets, $A \cap B$ is also an inductive set.
This set $A \cap B$ is one of the subsets
By the definition of $\omega$ it follows that $\omega \subseteq A \cap B$, and consequently that $\omega \subseteq B$.
$\blacksquare$
Sources
- 1960: Paul R. Halmos: Naive Set Theory ... (previous) ... (next): $\S 11$: Numbers