HP printer 4155e was working fine until 2 days ago when we got a message that we are connected to a wireless connection but not connected to the internet.

Here are a couple of points:

• A message that states you are connected to a wireless connection, but not to the internet is usually related to your router and restarting the router may solve that problem
• The Printer is an AirPrint printer, so when adding it you will want to select the AirPrint/Bonjour Printer Driver. No driver is needed from HP.
• To use AirPrint, the printer and the device needs to be connected to the same Wifi network. You will need to use the HP Easy Start software to connect the printer to the internet, since there are no controls on the printer to do that manually. You do not need to install any printer drivers using the Easy Start software, you just need to use it to connect the printer to your WiFi.

I suggest first restarting your router, then use the Easy Start software to connect the printer to your WiFi network. After that remove the driver from your computer and add it again using AirPrint/Bonjour.

Also, even without an active Internet (WAN) connection, AirPrint should work per se. The main thing is that the Wi-Fi router's local network (WLAN or LAN) is OK. A full power-cycle sequence (including the router, the printer and all connected computers/devices) can be tested as well. Is the printer possibly part of an ink subscription program, where the manufacturer may require an always-on Internet connection?