A description/proof of that for transitive set with partial ordering by membership, element is initial segment up to it
Topics
About: set
The table of contents of this article
Starting Context
- The reader knows a definition of transitive set.
- The reader knows a definition of initial segment up to element of set.
Target Context
- The reader will have a description and a proof of the proposition that for any transitive set with the at least partial ordering by membership (supposing that the ordering by membership is really a partial ordering), any element is the initial segment up to it.
Orientation
There is a list of definitions discussed so far in this site.
There is a list of propositions discussed so far in this site.
Main Body
1: Description
For any transitive set, \(S\), with the at least partial ordering by membership (supposing that the ordering by membership is really a partial ordering), any element, \(p \in S\), is the initial segment, \(seg \text{ }p\), up to \(p\), which is \(p = seg \text{ }p\).
2: Proof
For any element, \(p' \in seg \text{ }p\), of \(seg \text{ }p\), \(p' \in p\), because \(p' \lt p\) if and only if \(p' \in S\) and \(p' \in p\), by the definition of ordering by membership. For any element, \(p' \in p\), of \(p\), \(p' \in seg \text{ }p\), because \(p' \in S\) as \(S\) is transitive and so, \(p' \lt p\).
