Formula to count the number of different letters in a word

In sheets, suppose cell A1 contains the word CABBAGE.

I want to put a formula into, say, cell B1 that will count the number of different letters. So the output should be 5, because there are 5 different letters in CABBAGE.

I have a formula that will do this, however it is cumbersome and I bet there is a way to do it more concisely.

The formula that works is:

=if(isnumber(search("A",A1)),1,0) + if(isnumber(search("B",A1)),1,0) + ... + if(isnumber(search("Z",A1)),1,0)

The formula adds 1 each time a letter is present in the word. I'm just not sure of the best way to consolidate the 26 terms into something more manageable.