Containment of ACC$^0$ in TC$^0$
You can show directly how to simulate a polynomial fan-in modular gate using polynomially many threshold gates. The idea is that given inputs $x_1,\ldots,x_m$, you can use two threshold gates and an AND gate to determine whether $x_1 + \cdots + x_m = w$ for any $w$. Taking at most $m+1$ copies of this gadget and ORing them, we can simulate a single modular gate.