Yes, no idea what I was thinking when I typed that. I've used both extensively, in fact this message was sent over a PoE enabled WiFi AP.

For maximum comedy, I'm imagining running powerline ethernet to supply a network drop to a PoE switch, which then powers the AP.

I have literally done this many years ago. Not a Cisco 2960 or anything fancy, just a dumb 802.3af PoE switch.