Currently you are checking if the form has been posted:
But instead of posting the form, you are loading a new page.
$('#div1-wrapper').load(url + ' #div1');
So don't do that. Get rid of all the scripting - you don't need it for what you're doing here.
Instead of scripting, give the button a name and a value so that the empty() check will find something, and things will work as desired.
<button type="submit" name="btn_click" value="submitted" /> click me! </button>
Alternatively, you can get away with no value by using another technique to check if btn_click exists, by using filter_has_var, that lets you easily check for such things:
<button type="submit" name="btn_click" /> click me! </button>