in the case of sperm I can understand that there are so many of them that you can put a sample of semen under microscope and see a few of the cells, just randomly. But the egg is just a single big cell floating around in lots of liquid, right? So when they grab it together with the liquid, can they then isolate it? Indeed, can they even verify/prove the fact that it is there in the sample in the first place?
How does this procedure work?
Human egg cells are actually quite large. They can be seen with the naked eye, though a magnifying lens can also be useful.
That is straight up freaky.
Why?
Because I had imagined them to be much more tiny. (Damn my rudimentary grasp of biology)
Realizing that the videos about eggs and sperm are not to scale was something unexpected for me.
They’re the biggest cell we’ve got.
Here’s a picture of a sperm being injected into an egg.
Wikipedia describes the procedure:
Pointless nitpick ahead 
Not true. You have nerve axons that extend from your spine down to the tip of your toes. That makes these cells about 3 feet long!
I thought nerve cells were the biggest.
Pointless nitpick of a pointless nitpick: That makes the nerve cell “longest”, not “biggest”.
heh. Let me nitpick your pointless nitpick of my pointless nitpick
I submit that “biggest” is up to definition. Is it length? Then nerve axon wins. Diameter? Then the egg cell wins. Total cellular volume? I dunno - I’ll do some back 'o the envelope calculations and get back to you (my gut says “axon”). Ditto for total cellular surface area.
I’d put my money on a gut epithelial cell there - microvilli FTW.
Well, to actually answer the OP, Here is a description of the technique, along with an ultrasound picture of the follicles that the eggs develop in.
Egg harvesting is brutal. They use a cocktail of strong hormones over a couple months time to artificially ripen as many eggs as possible in your ovaries (usually 5 to 15 in a successful procedure I believe). Then they punch a hole through the wall of your vagina, and use tools to puncture the follicle of each ripe egg in the ovary and suck it out. There’s a lot of blood and fluid they take out along with the egg.
Just a aside, IIRC the largest single cell known is the ostrich egg.
If a nerve axon is one meter in length and has a diameter of one micron, while a human egg has a diameter of 100 microns (values from wikipedia), the cellular volume of the axon is 1.5 times larger than the egg.
(assuming the egg is spherical and the axon is a cylinder)
This is a little bit of an overstatement.
The hormones which induce egg production are given over a period of 10 days to 2 weeks or so, not a couple of months. Prior to that, different hormones may be given for a week or two to suppress native hormone production, not to artificially ripen eggs.
There certainly shouldn’t be a lot of blood taken out. Doctors do try to maximize the amount of fluid they remove (for example, by puncturing immature follicles) to decrease the risk of OHSS.
While it certainly is an unpleasant experience, I don’t think everyone who has undergone it would describe it as “brutal”. The egg retrieval itself is a minor outpatient procedure which entails less immediate pain and has a shorter recovery time than many dental procedures.
