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successor_set [2014/03/21 11:11] 127.0.0.1 external edit |
successor_set [2014/12/05 10:51] (current) nikolaj |
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==== Set ==== | ==== Set ==== | ||
| @#55CCEE: context | @#55CCEE: $X$ | | | @#55CCEE: context | @#55CCEE: $X$ | | ||
- | + | | @#FFBB00: definiendum | @#FFBB00: $ {\mathrm{succ}}\ X \equiv X \cup \{X\} $ | | |
- | | @#FFBB00: definiendum | @#FFBB00: $ S(X) \equiv X \cup \{X\} $ | | + | |
==== Discussion ==== | ==== Discussion ==== | ||
=== Theorems === | === Theorems === | ||
- | ^ $ X\in S(X) $ ^ | + | ^ $ X\in {\mathrm{succ}}\ X $ ^ |
- | ^ $ (S(X)=S(Y))\implies (X=Y) $ ^ | + | ^ $ ({\mathrm{succ}}\ X={\mathrm{succ}}\ Y)\Rightarrow (X=Y) $ ^ |
- | ^ $ (Y\in S(X))\iff (Y=X\lor Y=\{X\}) $ ^ | + | ^ $ (Y\in {\mathrm{succ}}\ X)\Leftrightarrow (Y=X\lor Y=\{X\}) $ ^ |
- | ^ $ X\neq S(X) $ ^ | + | ^ $ X\neq {\mathrm{succ}}\ X $ ^ |
=== Reference === | === Reference === | ||
Wikipedia: [[http://en.wikipedia.org/wiki/Successor_ordinal|Successor ordinal]] | Wikipedia: [[http://en.wikipedia.org/wiki/Successor_ordinal|Successor ordinal]] | ||
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ProofWiki: [[http://www.proofwiki.org/wiki/Definition:Successor_Set|Successor set]] | ProofWiki: [[http://www.proofwiki.org/wiki/Definition:Successor_Set|Successor set]] | ||
==== Parents ==== | ==== Parents ==== | ||
- | === Context === | + | === Requirements === |
[[Singleton]], [[Union]] | [[Singleton]], [[Union]] |