## Road to Zorn’s Lemma Part 3

Hello, infinite-set reader. Set theory is back.

As I said in the previous post, today we will discuss about the theorem of any poset that the every non-empty well ordered subset has the smallest bound on the poset. Happy reading.

**Theorem :**

Given any poset with the every non-empty well ordered subset has the smallest bound in . Given fuction with for all . For any then is well ordered set such that :

and if then apply :

**Proof :**

To prove the above theorem is sufficiently proved that because if applicable then by fuction definiton of . On the other hand , since then by

Theorem (ii)in this post applies . Since is then . Thus, we get .It will be proven that with . Given any with . Further taken any then by

Theorem (y)in this post can be consider . Furthermore, . Because of has the smallest element then also has the smallest element. Furthermore, . In other words, for any with then has the smallest element which resulted is well ordered set. Next, byTheorem (i)in this post then . Thus, we get and if then . In other words, is the smallest element of .It will then be proved that satisfy the -chain axioms. It has been proved that is well ordered set. According to the hypothesis on the proof of this theorem, then has the largest element. Let’s say . Defined . Will be shown is -chain. Since is well ordered set then also well ordered set with is the smallest element and is the largest element.

Condition (i)of definition -chain is fulfilled.Next, take any with then . Since then . Note that and are the last two elements on -chain . The result is valid for all .

Condition (ii)of definition -chain is fulfilled.Next, take any with then is the upper bound . There are two possibilities. The first possibility, if there is no such that for all then . Second possibility, if there is such that for all then .

Condition (iii)of definition -chain is fulfilled. In other words is a -chain with being its largest element.Since is the set of all elements -chain then . Then we get is -chain .

You should reopen my previous post to get a good understanding. This is because my posts are related to each other. If you forget a little definition or theorem, you will be confused. I try to make this discussion easier by providing links. I hope this makes it easier for you to understand this discussion.

Thanks for reading. May be useful.

## Road To Zorn’s Lemma Part 2

Hello, infinite-set reader. Set theory is back.

Before reading this post, you should read the previous post first. It deals with the definition of and . As a reminder only if you forget. The post is still about the character of and . Happy reading.

**(x) Theorem :**

If then either of the following two applies, or .

**Proof :**

Defined then is well ordered set because is subset of and both of which are well ordered set. Further, by definition -chain

condition (i)then and by definition -chaincondition (iii)then are in and also in which resulted . Next, assume then . if then by definition -chaincondition (ii). Contradictions with . In other words, if then which resulted . Because of then by definition -chaincondition (i)we get . Even though and are two things that are contradictory then the assumption is rejected. Theorem is proven.

**(y) Theorem :**

If then there is a unique such that .

**Proof :** (more…)

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