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I am working on a proof that involves the Ordered Pair Property and Ordered Triples and the notation used in the proof is $\langle x,\langle y,z\rangle\rangle$.

Can this ordered triple notation also be written as $\{\{x\},\{\{y\},\{y,z\}\}\}$? I have yet to see that notation used.

Thank you!

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What is Fundamental Theorem of Ordered Pairs - it doesn't come up on Google ? –  Tom Collinge Jun 23 at 15:05
    
Sorry. I meant the Ordered Pair Property. I will make the edit. –  user42864 Jun 23 at 15:06

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If $\langle u, v \rangle = \{ \{ u \}, \{ u,v \} \}$, then your formula is wrong:

$$ \langle x, \langle y, z \rangle \rangle = \{ \{ x \}, \{ x, \{\{y\}, \{y,z\}\}\} $$

If you're simply conjecturing a way to realize ordered triples, your formula isn't good:

$$ \langle \{ s \}, t, t \rangle = \{ \{ \{ s \}\}, \{\{t\}\} \} $$ $$ \langle \{ t \}, s, s \rangle = \{ \{ \{ t \}\}, \{\{s\}\} \} $$

The left hand sides are different, but the right hand sides are the same set.

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Ok. I see the error. I had a feeling my notation was incorrect. Thank you for your help. –  user42864 Jun 23 at 15:18

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