You can do this using the Stash addon (https://github.com/croxton/Stash).
The basic idea will be that you create a Stash variable in your first EE tag, and then use that variable to exclude those entries by ID in the second entries tag.
Here is a rough idea with untested code:
Your slider code:
{exp:channel:entries
channel="whatevs"
limit="4"
}
<!-- Create a list of entry IDs that you are using. i.e. 1|2|3|4 -->
{exp:stash:append name="entries_ids_to_exclude"}{if count != 1}|{/if}{entry_id}{/exp:stash:append}
<!-- Your slider code -->
{/exp:channel:entries}
Your other listing:
{exp:channel:entries
channel="whatevs"
entry_id="not {exp:stash:get name='entries_ids_to_exclude'}"
}
<!-- All the other entries you want to output excluding the ones above -->
{/exp:channel:entries}
As I write this, I am wondering if this will work. You may need to include the second entries tag in an {embed} tag to pass the stash variable to it. So, the second part may need to be an embed:
{embed="template/template_group" entries_to_exclude="{exp:stash:get name='entries_ids_to_exclude'}"}
And then the code in your embed template would be:
{exp:channel:entries
channel="whatevs"
entry_id="not {embed:entries_to_exclude}"
}
<!-- All the other entries you want to output excluding the ones above -->
{/exp:channel:entries}