My opponent has 8 cards in their trash. Can I use this card's [Start of Your Main Phase] effect to trash the top 2 cards of their deck, then process the part of the effect after "then"?

My opponent has 11 cards in their trash. This card's [Start of Your Main Phase] effect doesn't trash the top 2 cards of their deck, but can I process the part of the effect after "then"?