As far as I can tell, the books don't explicitly state that they are for 5th edition.
However, there's a phrase on the back that says:
When you're ready for even more, expand your adventures with the fifth edition other core rulebook #1 and other core rulebook #2.
That's at least an indirect hint that the book is for 5e.
Volo's Guide To Monsters (and probably also the other non-core-rulebooks, but I don't have those atm) states on the back cover:
For use with the fifth edition Player's Handbook, Monster Manual, and Dungeon Master's Guide.
There is also one tiny, indirect hint in the Dungeon Master's Guide, p. 11, that indicates that the book is for an edition past 4e (which, currently, only leaves 5e afaik):
The pantheon of the Dawn War is an example of a
pantheon assembled from mostly preexisting elements
to suit the needs of a particular campaign. This is
the default pantheon in the fourth edition Player's
Handbook (2008). The pantheon is summarized in the
Dawn War Deities table. [...]
I couldn't find anything else in the PHB or MM.
On a side note, I recommend checking out this question: How do I know which edition of Dungeons and Dragons (D&D) the books I'm looking at are for?.
It's not an obvious indication since you need to know which logo equals which edition, but if you find a random D&D book that doesn't specify its edition, you can compare the logos.