ANY (or its synonym SOME) is a syntax sugar for EXISTS with a simple correlation:
SELECT *
FROM mytable
WHERE x <= ANY
(
SELECT y
FROM othertable
)
is the same as:
SELECT *
FROM mytable m
WHERE EXISTS
(
SELECT NULL
FROM othertable o
WHERE m.x <= o.y
)
With the equality condition on a not-nullable field, it becomes similar to IN.
All major databases, including SQL Server, MySQL and PostgreSQL, support this keyword.