Definition:Slowly Well-Ordered Class under Subset Relation

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Definition

Let $N$ be a class.

Let $N$ be well-ordered under the subset relation such that the following $3$ conditions hold:

\((\text S_1)\)   $:$   $\O \in N$ is the smallest element of $N$      
\((\text S_2)\)   $:$   Each immediate successor element contains exactly $1$ more element than its immediate predecessor      
\((\text S_3)\)   $:$   Each limit element $x$ is the union of its lower section $\bigcup x^\subset$      


Then $N$ is slowly well-ordered under the subset relation.


Also known as

Some sources do not hyphenate, and have well ordered for well-ordered.

Some sources refer to under inclusion for under the subset relation.


Sources