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If only the first mode is chosen, or if both modes are chosen, and the chosen target is illegal as the ability tries to resolve, the ability won't resolve and none of its effects will happen.

Prophetic Titan's ability checks the number of card types in your graveyard as it's put on the stack. If you have four or more types in your graveyard at that time, but fewer than four as it resolves, you will still get to use both modes chosen.

The card types that can appear in a graveyard are artifact, battle, creature, enchantment, instant, kindred, land, planeswalker, and sorcery. Legendary, basic, and snow are supertypes, not card types; Giant and Wizard are subtypes, not card types.