Set whose every Finite Subset is Nest is also Nest

Theorem

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Let $x$ be a set of sets with the property that:

every finite subset of $x$ is a nest.

Then $x$ is a nest.

Proof

Let $x$ be a set of sets with the given property.

Let $a, b \in x$ be arbitrary.

Then:

$\set {a, b} \subseteq x$

and so $\set {a, b}$ is a nest.

That is:

$a \subseteq b$ or $b \subseteq a$

As $a$ and $b$ are arbitrary, it follows that:

$\forall a, b \in x: a \subseteq b$ or $b \subseteq a$

This article, or a section of it, needs explaining. In particular: As per definition, it seems to me that the above line already establishes the definition of nest? What does Subset Relation is Ordering add? You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by explaining it. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Explain}} from the code.

The result follows from Subset Relation is Ordering.

$\blacksquare$

Sources

• 2010: Raymond M. Smullyan and Melvin Fitting: Set Theory and the Continuum Problem (revised ed.) ... (previous) ... (next): Chapter $4$: Superinduction, Well Ordering and Choice: Part $\text {II}$ -- Maximal principles: $\S 5$ Maximal principles