Trying to make efficient "all vendors who can provide all items"

My mind is drawing a blank. Please consider:
TABLE 1: items: list of items in catalog
item_id | item_description

TABLE 2: vendors: list of vendors who provide 1 or more items
vendor_id | vendor_name

TABLE 3: item_vendors: record existence indicates vendor can provide item
item_id | vendor_id

QUESTION:
I have a list of say 5 items, and I want to find all vendors who can provide
ALL 5 items

Solution 1:
SELECT vendor_id
FROM vendors
WHERE EXISTS (SELECT 1 FROM item_vendors WHERE vendors.vendor_id =
item_vendors.vendor_id AND item_id = 'item_1')
AND EXISTS (SELECT 1 FROM item_vendors WHERE vendors.vendor_id =
item_vendors.vendor_id AND item_id = 'item_2')
AND EXISTS (SELECT 1 FROM item_vendors WHERE vendors.vendor_id =
item_vendors.vendor_id AND item_id = 'item_3')
AND EXISTS (SELECT 1 FROM item_vendors WHERE vendors.vendor_id =
item_vendors.vendor_id AND item_id = 'item_4')
AND EXISTS (SELECT 1 FROM item_vendors WHERE vendors.vendor_id =
item_vendors.vendor_id AND item_id = 'item_5')

Solution 2:
SELECT vendors.vendor_id
FROM vendors, items AS item_1, items AS item_2, items AS item_3, items AS
item_4, items AS item_5
WHERE items_1.vendor_id = vendors.vendor_id AND items_1.item_id = 'item_1'
AND items_2.vendor_id = vendors.vendor_id AND items_2.item_id = 'item_2'
AND items_3.vendor_id = vendors.vendor_id AND items_2.item_id = 'item_3'
AND items_4.vendor_id = vendors.vendor_id AND items_2.item_id = 'item_4'
AND items_5.vendor_id = vendors.vendor_id AND items_2.item_id = 'item_5'

Yep, both my solutions are pretty ugly, especially in situations where my
list of items that need to be provided grow large.

There must be a better way. Can anyone help me with this?

Thanks

Terry Fielder
Manager Software Development and Deployment
Great Gulf Homes / Ashton Woods Homes
terry(at)greatgulfhomes(dot)com
Fax: (416) 441-9085