Intersection of Inductive Set as Subset of Real Numbers is Inductive Set

From ProofWiki
Jump to navigation Jump to search

Theorem

Let $\AA$ be a set of inductive sets defined as subsets of real numbers.

Then their intersection is an inductive set.


Proof

By definition of inductive set:

$\forall S \in \AA: 1 \in S$

Thus by definition of set intersection:

$\ds 1 \in \bigcap_{S \mathop \in \AA} S$


Also by definition of inductive set:

$\forall S \in \AA: x \in S \implies x + 1 \in S$

So:

\(\ds x\) \(\in\) \(\ds \bigcap_{S \mathop \in \AA} S\)
\(\ds \leadsto \ \ \) \(\ds \forall S \in \AA: \, \) \(\ds x\) \(\in\) \(\ds S\) Definition of Intersection of Set of Sets
\(\ds \leadsto \ \ \) \(\ds \forall S \in \AA: \, \) \(\ds x + 1\) \(\in\) \(\ds S\) Definition of Inductive Set as Subset of Real Numbers
\(\ds \leadsto \ \ \) \(\ds x + 1\) \(\in\) \(\ds \bigcap_{S \mathop \in \AA} S\) Definition of Intersection of Set of Sets

Hence the result, by definition of inductive set.

$\blacksquare$


Sources