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[parent] $(a,b)=(c,d)$ if and only if $a=c$ and $b=d$ (Proof)

Following is a proof that the ordered pairs $ (a,b)$ and $ (c,d)$ are equal if and only if $ a=c$ and $ b=d$.

Proof. If $ a=c$ and $ b=d$, then $ (a,b)=\{\{a\},\{a,b\}\}=\{\{c\},\{c,d\}\}=(c,d)$.

Assume that $ (a,b)=(c,d)$ and $ a=b$. Then $ \{\{c\},\{c,d\}\}=(c,d)=(a,b)=\{\{a\},\{a,b\}\}=\{\{a\},\{a,a\}\}=\{\{a\},\{a\}\}=\{\{a\}\}$. Thus, $ \{c,d\}\in\{\{a\}\}$. Therefore, $ \{c,d\}=\{a\}$. Hence, $ a=c$ and $ a=d$. Since it was also assumed that $ a=b$, it follows that $ a=c$ and $ b=d$.

Finally, assume that $ (a,b)=(c,d)$ and $ a \neq b$. Then $ \{a\} \neq \{a,b\}$. Note that $ \{\{a\},\{a,b\}\}=(a,b)=(c,d)=\{\{c\},\{c,d\}\}$. Thus, $ \{c\} \in \{\{a\},\{a,b\}\}$. It cannot be the case that $ \{c\}=\{a,b\}$ (lest $ a=c=b$). Thus, $ \{c\}=\{a\}$. Therefore, $ a=c$. Hence, $ \{\{a\},\{a,b\}\}=\{\{c\},\{c,d\}\}=\{\{a\},\{a,d\}\}$. Note that $ \{a,b\} \in \{\{a\},\{a,d\}\}$. Since $ \{a\} \neq \{a,b\}$, it must be the case that $ \{a,b\}=\{a,d\}$. Thus, $ b \in \{a,d\}$. Since $ a \neq b$, it must be the case that $ b=d$. It follows that $ a=c$ and $ b=d$. $ \qedsymbol$



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Cross-references: ordered pairs, proof

This is version 6 of $(a,b)=(c,d)$ if and only if $a=c$ and $b=d$, born on 2006-09-07, modified 2006-10-09.
Object id is 8320, canonical name is AbcdIfAndOnlyIfAcAndBd.
Accessed 1018 times total.

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AMS MSC: 03-00 (Mathematical logic and foundations :: General reference works )
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