I was reading Hodel's otherwise excellent An Introduction to Mathematical Logic and, in the appendix on number theory, specifically on the section on congruences, he seems to make a slip. Let $a, b, m \in \mathbb{Z}$ and $m > 0$. He claims that:
If $a \equiv b \mod m$ and $0 \leq a$ and $b < m$, $a=b$
But that seems clearly mistaken, for take $a = 22$, $b= 4$ and $m = 9$. In that case, clearly $22 \equiv 4 \mod 9$, $0 \leq 22$, and $4 < 9$, but equally clearly $22 \not = 4$. Am I missing something? If not, and this is indeed a (trivial) counter-example to Hodel's claim, is there any other similar property about congruences that he could be referring to?
I think it is just unfortunate notation. The statement should be read $$ 0\leq \text {($a $ and $b $)}<m, $$ and not $$ 0\leq a, \text { and } b <m. $$