> What I'm trying to achieve is the element with id="save-search" only 
> gets hidden if it does not contain a div with the class="error-message"

How about this? 

$("#save-search:not([div.error-message])").hide();

It actually reads very much like the sentence above, once you know what all
the syntax means:   

 #save-search            // select an element with id=save-search
   :not(                      // as long as the following is not found:
     [                         // descendent elements containing
      div                     // a div
       .error-message    // with class error-message
     ])

Notice that it's _very_ important to put the :not right after the id name.
If there was a space after the id it would select _descendants_ of
#save-search that do not contain div.error-message. 


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