2 Incorporated information from the comments and an and added info about special actions. If we're gonna talk about effects that don't use the stack better cover all of them
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You opponent was correct and the creature will die. 

In Magic the Gathering there is no such thing as a abilityvirtually nothing that "triggers instantaneously". With the exception of activated abilities that only create mana all abilities go on the stack. AfterThere are a few special actions that a player can perform that do not use the stack as well, but these are not considered "abilities" by the games rules. Also if there is an effect that says to do something "instead of" something else then the new action will occur when the original would and not go on the stack.

None of these exceptions are relevant here so when combat damage happens the the trigger goeswill go on the stack, then state based actions will be checked and the creature will die before the ability resolves or any player gains priority. Even if you had a special action you could perform that would save the creature it will be dead before you have a chance to do it.

You opponent was correct. In Magic the Gathering there is no such thing as a ability that "triggers instantaneously". With the exception of activated abilities that only create mana all abilities go on the stack. After the trigger goes on the stack state based actions will be checked and the creature will die before the ability resolves.

You opponent was correct and the creature will die. 

In Magic the Gathering there is virtually nothing that "triggers instantaneously". With the exception of abilities that only create mana all abilities go on the stack. There are a few special actions that a player can perform that do not use the stack as well, but these are not considered "abilities" by the games rules. Also if there is an effect that says to do something "instead of" something else then the new action will occur when the original would and not go on the stack.

None of these exceptions are relevant here so when combat damage happens the the trigger will go on the stack, then state based actions will be checked and the creature will die before the ability resolves or any player gains priority. Even if you had a special action you could perform that would save the creature it will be dead before you have a chance to do it.

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You opponent was correct. In Magic the Gathering there is no such thing as a ability that "triggers instantaneously". With the exception of activated abilities that only create mana all abilities go on the stack. After the trigger goes on the stack state based actions will be checked and the creature will die before the ability resolves.