In the article Characterization of wavelets and MRA wavelets on local fields of positive characteristic, in page 11, As seen in the picture, The author refers to 16(Multiresolution analysis on local fields) for his statement. But in 16, "theorem 3", it seems does not say that.
In my opinion, the statement is true if "theorem 3" is stated in the form "if, and only if".
Theorem 3 in 16(Multiresolution analysis on local fields), page 8:
