JQuery UI not loading without explicit loading of jQuery

jQuery and the jQuery UI Slider are both registered by default by WordPress. In order to use them, all you should need to do is something like this:

add_action( 'wp_enqueue_scripts', 'wpse192997_load_slider' );
function wpse192997_load_slider() {
    wp_enqueue_script( 'jquery' );
    wp_enqueue_script( 'jquery-ui-slider' );
}

Note, though, that WordPress uses jQuery in noConflict mode, which means that the $ variable used in most jQuery code samples on the Internet won’t be available. You can get around that by using the techniques listed in the jQuery noConflict Wrappers section of the Codex article on wp_enqueue_script().