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This non-implication, Form 324 \( \not \Rightarrow \) Form 411, whose code is 6, is constructed around a proven non-implication as follows:

• An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 9967, whose string of implications is: 322 \(\Rightarrow\) 324
• A proven non-implication whose code is 5. In this case, it's Code 3: 641, Form 322 \( \not \Rightarrow \) Form 10 whose summary information is:
  

  Hypothesis	Statement
  Form 322	<p> \(KW(WO,\infty)\), <strong>The Kinna-Wagner Selection Principle for a well ordered family of sets:</strong> For every  well ordered set \(M\) there is a function \(f\) such that for all \(A\in M\), if \(|A|>1\) then \(\emptyset\neq f(A)\subsetneq A\).  (See <a href="/form-classes/howard-rubin-15">Form 15</a>). </p>

  
  

  Conclusion	Statement
  Form 10	<p> \(C(\aleph_{0},< \aleph_{0})\):  Every denumerable family of non-empty finite sets has a choice function. </p>

  
  
• An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 9570, whose string of implications is: 411 \(\Rightarrow\) 412 \(\Rightarrow\) 10

The conclusion Form 324 \( \not \Rightarrow \) Form 411 then follows.

Finally, the
List of models where hypothesis is true and the conclusion is false:

Name	Statement
\(\cal N50(E)\) Brunner's Model III	\(E\) is a finite set of prime numbers.For each \(p\in E\) and \(n\in\omega\), let \(A_{p,n}\) be a set of atoms ofcardinality \(p^n\)