MHB Proof of Second Sentence:$m<n \leftrightarrow m'<n'$

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The discussion centers on proving the logical equivalences involving natural numbers, particularly focusing on the implications of the second sentence: $m<n \leftrightarrow m'<n'$. The user attempts to establish this relationship through various set operations and logical deductions, ultimately questioning how to handle the case when $m=n$. They argue that if $m=n$, then $m'$ must equal $n'$, which contradicts the assertion that $m'<n'$. Additionally, they explore the fourth proposition regarding the relationship between $m \leq n'$ and $m \leq n \lor m=n'$. The conversation highlights the complexities of proving these mathematical statements and the necessity of addressing edge cases in the proofs.
evinda
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Hello! (Wave)

I am looking at the following sentence:

For any natural numbers $m,n$ it holds:

  • $m \leq n \leftrightarrow m' \leq n'$
  • $m<n \leftrightarrow m'<n'$
  • $m<n' \leftrightarrow m \leq n$
  • $m \leq n' \leftrightarrow m \leq n \lor m=n'$

I tried to prove the second sentence like that:

$$m<n \rightarrow m \in n \rightarrow m \subset n \wedge \{m\} \subset n \rightarrow m \cup \{ m \} \subset n \rightarrow m \cup \{m \}=n \lor m \cup \{ m \} \in n$$

From the relation $m \cup \{m \} \in n$ we get that $m \cup \{ m \} \in n \cup \{ n \} \rightarrow m' \in n' \rightarrow m'<n'$.

$$m'<n' \rightarrow m' \in n' \rightarrow m \cup \{ m \} \in n \cup \{ n \} $$

$$m \in m \cup \{ m \} \rightarrow m \in n \cup \{ n \} \rightarrow m \subset n \cup \{ n \} \rightarrow m \subset n \lor m \subset \{ n \}$$

From the relation $m \subset n$ we conclude that $m \in n \lor m=n$.

How could we reject the case $m=n$ ? (Thinking)
 
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evinda said:
$$m'<n' \rightarrow m' \in n' \rightarrow m \cup \{ m \} \in n \cup \{ n \} $$

$$m \in m \cup \{ m \} \rightarrow m \in n \cup \{ n \} \rightarrow m \subset n \cup \{ n \} \rightarrow m \subset n \lor m \subset \{ n \}$$

From the relation $m \subset n$ we conclude that $m \in n \lor m=n$.

How could we reject the case $m=n$ ?
You could argue that if $m=n$, then $m'=n'$ and therefore $m'<n'$, i.e., $m'\in n'$, is impossible.
 
Evgeny.Makarov said:
You could argue that if $m=n$, then $m'=n'$ and therefore $m'<n'$, i.e., $m'\in n'$, is impossible.

I see! (Nod)

Could we show the fourth proposition ($m \leq n' \leftrightarrow m \leq n \lor m=n'$) like that? (Thinking)

$$m \leq n' \leftrightarrow m=n' \lor m \in n' \leftrightarrow m=n' \lor m \in n \cup \{ n \} \leftrightarrow m=n' \lor m \subset n \cup \{ n \} \\ \leftrightarrow m=n' \lor m \subset n \lor m \subset \{ n \} \leftrightarrow m=n' \lor m \leq n \lor m \subset \{ n \}$$

It holds that:

$$m \subset \{ n \} \rightarrow m=\{ n \} \lor m \in \{ n \} \rightarrow m=\{ n \} \lor m=n$$

But we know the following:

$$m \leq n \leftrightarrow m \in n \lor m=n$$

and:

$$\{ n \} \subset n$$

So, $m \subset \{ m \} \rightarrow m \leq n$.

Therefore,

$$m \leq n' \leftrightarrow m=n' \lor m \leq n.$$
 
evinda said:
Hello! (Wave)

I am looking at the following sentence:

For any natural numbers $m,n$ it holds:

  • $m \leq n \leftrightarrow m' \leq n'$

I tried to prove the second sentence like that:

$$m<n \rightarrow m \in n \rightarrow m \subset n \wedge \{m\} \subset n \rightarrow m \cup \{ m \} \subset n \rightarrow m \cup \{m \}=n \lor m \cup \{ m \} \in n$$

From the relation $m \cup \{m \} \in n$ we get that $m \cup \{ m \} \in n \cup \{ n \} \rightarrow m' \in n' \rightarrow m'<n'$.

We say that $m \leq n \leftrightarrow m \in n \lor m=n$ but then we only use the case when $m \in n$.. Do we also have to conclude something from $m=n$? :confused:
 

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