Opp: Summons TIV in faceup attack mode.
Me: Has set Book of Moon.
Opp: Use TIV effect, discards a card from their to activate effect.
Me: Chain, use BoM on TIV in reponse to effect.

Would the effect still go through? Also, how can you tell if a monster has priority? Is it if the monster has an activation effect, or is it if the effect has an activation cost only?

Monsters don't have priority. The Player does.

When a monster with a triggered Ignition effect (LP, Discard, no battle, tribute monsters, etc) is summoned, that player has the RIGHT, as the turn, player to activate it.

The monster's effect still goes through because it was paid for.



Waldo10032 – The Fiendish Duelist says:
“Hang with male chauvinist pigs and perverts, who point water pistols and women and squirt shirts. Been a bad boy since diapers and Gerberts. My first words were ‘bleep, bleep’ and ‘curse, curse’”

Unless it says the monster has to be face up on the field during the resolution, then it will STILL resolve, even if the monster is flipped face down.

If you want to be able to stop effects without Divine Wrath there is....

Skill Drain (only stops face up mons effects, but flip effects won't go either since they flip face up)

Royal Command (for Flip effects)

Or get around their activation requirements like, for example.

Banisher of the Light to stop Witch, Sangan, Tomato, Giant Rat, etc from going to the graveyard

Card effects that SEND to the graveyard if they need to be DESTROYED.

to add to what waldo already said, priority basically works like this.

when something happens, be it a summon, atk, set, or anything else that isn't chainable, both players are given a chance to "respond" to what just occured.

first the turn player is given a chance to respond (this is where priority really sits, turn player priority allows the turn player to go first)

the turn player can respond with any speed 2 or 3 effect. also if responding to a summon, the turn player can use any manually activated effect of the monster he just summoned (as long as any requirments have been met of course).

then the opponent is given a chance to respond, and can do so with any speed 2 or 3 effect, if the turn player responded already, the opponents effect will chain to the turn player's effect.

so, as for which monsters have priority (which means which monsters can be used with priority), just ask yourself, "Can i activate this effect during my main phase, and does it have any activation requirments preventing me from activating it at this time?" and you'll know if you can activate its effect.