Can I use the printer wirelessly without a router?

Yes, you can use Wi-Fi Direct to print wirelessly without a router. Make sure Wi-Fi Direct is enabled, and to check its status, network name, and password, touch the Wi-Fi Direct icon on the printer control panel. Learn more at www.hp.com/go/wifidirectprinting. (Page 2)

From the official HP Deskjet Ink Advantage 5000 All-in-One Printer manual, page 2

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