Usually 0!=1 is not deducted and can not be proven - its simply defined. Not arbitrarily of course but in way so it "fits" the rules for the rest of the factorials (similar applies to a^0=1 or a^(-n)=1/a^n).
And this definition fits in different ways, one of which you had shown. For every integer n>1 it is true that (n-1)! = n!/n. If you apply that to n=1, too you arrive at 0! Better not apply it to n=0, though.There are other ways for a logical continuation or you may arrive a n! using the Gamma function.