Apple - How to start Apple Hardware Test without Apple Software Disk (MBP13 mid-2010)?

By knowing your exact Model Identifier (MacBokPro?,?) you could look in to this data base for the correct version of AHT to download.

Examples:

MacBookPro1,1 3A106
MacBookPro1,1 3A107
MacBookPro2,1 Mac-F42189C8
MacBookPro2,2 Mac-F42187C8
MacBookPro3,1 Mac-F42388C8
MacBookPro3,1 Mac-F4238BC8
MacBookPro4,1 Mac-F42C86C8
MacBookPro4,1 Mac-F42C89C8
MacBookPro5,1 Mac-F42D86A9
MacBookPro5,1 Mac-F42D86C8
MacBookPro5,2 Mac-F2268EC8
MacBookPro5,3 Mac-F22587C8
MacBookPro5,4 Mac-F22587A1
MacBookPro5,5 Mac-F2268AC8
MacBookPro6,1 Mac-F22589C8
MacBookPro6,2 Mac-F22586C8
MacBookPro7,1 Mac-F222BEC8
MacBookPro8,1 Mac-94245B3640C91C81
MacBookPro8,2 Mac-94245A3940C91C80
MacBookPro8,3 Mac-942459F5819B171B
MacBookPro9,1 Mac-4B7AC7E43945597E
MacBookPro9,2 Mac-6F01561E16C75D06
MacBookPro10,1 Mac-C3EC7CD22292981F

Just to answer the question of how to get AHT and assuming your Mac is indeed a 13inch mid-2010 MacBook Pro your model is MacBookPro7,1 and for that model you can download AHT from Apple with the link below and follow my steps below to put it on a flash drive.

Apple AHT for MacBookPro7,1 - 022-4653-A.dmg - do not use unless you've verified you indeed have a MacBookPro7,1, it won't mess up your computer if it isn't, it just won't boot.

To some these steps may not make sense, I'm just reformatting what I was told and worked when I did this for my iMac to a guide for this specific MacBook Pro, I don't dare wipe my flash drive to try another method as I don't want to loose its power. Continuing on...

(your flash drive will be WIPED)
1. Insert your flash drive of at least 16MB capacity and open Disk Utility then select your flash drive on the left and go to the Partition tab.
2. Select 1 Partition in the drop down then go to Options... and select GUID Partition Table then verify Format is set to Mac OS Extended (Journaled)
3. Finally enter a name for your drive (I recommend something simple like AHT-FD, do not use spaces) and click Apply
4. After formatting is finished mount (double click) the 022-4653-A.dmg you downloaded and copy everything to your freshly formatted AHT-FD flash drive.
5. In Finder open Go in the menu bar and select Go to Folder... then paste /Volumes/AHT-FD/System/Library/CoreServices/.diagnostics/ and click Go
6. Copy the diags.efi file and paste it to the root of your AHT-FD flash drive.
7. For the final step open Terminal and paste this code then enter your password, your computer will immediately reboot after the command completes, save any work first

cd /Volumes/AHT-FD/ && sudo bless --mount /Volumes/AHT-FD --setBoot --file diags.efi && sudo reboot  

You should now be booting into AHT...
After you are finished shut down your computer, remove the flash drive, and at the start up chhime hold down Option on your keyboard and select your Mac's hard drive again.

Source for all this information: Here and here.