How come $REC$ languages are included in $NRE$ How come $REC$ languages are included in $NRE$.

The definition of a language in $REC$ is . $L \in REC$ if there is a turing machine $M$ that recognizes the language $L$ and that stops on any given input.

It makes sense that $REC$ is included in $RE$; 

$L \in RE$ if there is a turing machine $M$ that recognizes the language $L$ but may stop on some inputs;

The inclusion still makes sense right now.  But when we go to the $NRE$ i find it hard to understand;

$L \in NRE$ if there DOES NOT EXIST ANY turing machine $M$ that can recognize the language.

How can a language in $REC$ be in $NRE$ if $REC$ assumes that such a language would have a turing machine that stops on every intput yet a language that is $NRE$ must not have any turing machine that can recognize it?

Could the given definition of $NRE$ be wrong?
An explanation would be welcomed!
I've never seen "NRE" defined to mean "not recursively enumerable", which is the short way of saying what you've defined.  If you take it to mean "not recursively enumerable" then no recursive language is in NRE because every recursive language is recursively enumerable.

The only definition of "NRE" that I've seen is that people in the field of DNA and membrane computing seem to use it for the set of Turing-computable sets of natural numbers, i.e., the sets that can be written as $\{|x|\mid x\in L\}$ for some RE language&nbsp;$L$.