dnd 5e – Does a spell with verbal components that is cast because of Contingency take effect in a Silence spell?

I GM a campaign where the main villain is a 20th level wizard called EM (short for Evil Mage), with the spells contingency and dimension door. EM has a contingency spell, set to go off when xe drops to below 1/3 of xer max HP, with dimension door as the contingent spell (teleporting xem 450 feet straight up, although the destination really doesn’t matter).

Today, one of the players cast silence on EM, preventing xem from casting spells (they were in a space where EM couldn’t move out of the silenced area), and eventually the other players brought EM to 1/3 of xer max HP.

However, when I said that xe flashed away, the player argued with me, saying that the silence spell prevented casting spells with a verbal component, and because dimension door has a verbal component, it can’t be cast in the silenced area, even if it was cast by contingency. I said that no, the spell was already cast when the contingency was cast, so it didn’t need verbal components and silence didn’t prevent it from going off.

Does a silence spell prevent a spell with verbal components cast through contingency from going off?