Is the flash storage firmware update 1.0 for my MBA installed? How can I verify it?
Firmware update and FileVault
A forum post suggests that Firmware updates require access to /System/... in addition to the EFI partition (disk0s1).
Since /System/... is located on the startup volume it is encrypted if you have FileVault 2 enabled - due to Full Disk Encryption.
If you search further down in /System/... you will find the Revision number mentioned in the Apple knowledge-base document for this Firmware updateTPVABBF0:
/System/Library/CoreServices/Firmware Updates/HardDriveUpdate3/HDFW/APPLE_SSD_TS128E TPVABBF0

However, this alone is not a guarantee that the Firmware update has been installed. Before I had successfully applied the update, the contents already looked like this:

How to install...
Since there is no way around the encryption, I disabled FileVault - applied the update - and re-enabled FileVault.
You will notice that the update is installing if you see a grey bar during the boot process above which is a standard hard drive icon.
After the update, all the folders in /System/Library/CoreServices/Firmware Updates/... as you can see in the picture are still present and the Boot ROM Version: MBA51.00EF.B02 and SMC Version (system): 2.5f7 have not changed either.
While the Mac App Store still has the red badge for &quot;new updates&quot; it does not offer anymore to install the firmware update. So, I guess the firmware update has been installed successfully...