Phyrexian Tyranny

Enchantment
Whenever a player draws a card, that player loses 2 life unless he or she pays

.
Incendiary Command

Sorcery.
Choose two — Incendiary Command deals 4 damage to target player; or Incendiary Command deals 2 damage to each creature; or destroy target nonbasic land; or each player discards all the cards in his or her hand, then draws that many cards.
It is Alice's turn. She is on 10 life, has 6 cards in hand, and controls six untapped mana sources of various colours and two copies of Phyrexian Tyranny in play.
Bob is on 14 life, has 6 cards in hand, and controls no untapped mana sources.
Alice taps five mana to cast Incendiary Command, choosing the '4 damage to target player' and 'each player discards all the cards in his or her hand, then draws that many cards' modes.
The Phyrexian Tyranny triggers produced by this spelll are sufficient to cause enough life loss to kill both players. What is the result of the game?
This is what I suspect.
When Incendiary Command has finished resolving, 10 Phyrexian Tyranny triggers go on the stack for Alice and 12 for Bob. They can't all go off at once, so we use APNAP order. Alice is the active player, and so her triggers are put on the stack first. Then, Bob's triggers are put on the stack. When everything is on the stack and both players pass, the triggers start to resolve. Bob's triggers went on the stack first and so will resolve first, and because he doesn't have any untapped mana, he loses two life for each trigger. There are enough triggers to reduce his life total to below zero, and when they do he will lose the game as a state-based effect before Alice's triggers start to resolve. Therefore, the game is over and Alice wins.
Am I correct?