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Is there a logical symbol for "is of the form" which can be used as a shorthand in a statement like:

"Any even natural number is of the form $2n$"

?

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    Not as far as I know: "is of the form" is plenty short. One of its traditional uses in number theory has largely been replaced by the congruence notation. Anyway, we should concentrate more on being understood than on being brief.2011-07-20

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Something like "If $x$ is an even natural number then $\exists n \in \mathbb{N} : x=2n$"?