Media for NAT router, AV and firewall

NAT router, AV and firewall

If I have a NAT router that stops incoming packets (grc.com says my machine is completly stealthed even when I use only the NAT firewall) and a good updated AV-software that will detect virus and trojans...

Is there then any reason at all to have a software firewall?

I have always had one, to monitor programs, but it takes both RAM and CPU% and my AV should detect any trojans before they send anything, right?

My AV both has a normal program-scan (scans all started programs), a web-scanner (scans everything downloaded with a browser) and a mail-scanner (scans everything received through mail).

It has a "network scanner" as well which I don't use (think that is supposed to scan network traffic, sound a bit like a firewall :-)