Is array all NULLs in PostgreSQL

1 = ALL(arr) IS NULL AND 2 = ALL(arr) IS NULL

1 and 2 can be any two distinct numbers.

Alternatives and performance

There are many ways. I assembled a quick test case:

SELECT arr::text     , -1 = ALL(arr) IS NULL                              AS xsimple     , 1 = ALL(arr) IS NULL AND 2 = ALL(arr) IS NULL      AS simple     , array_remove(arr, NULL) = '{}'                     AS array_rem     , cardinality(array_positions(arr, NULL))     = cardinality(arr)                                   AS array_pos     , TRUE = ALL (SELECT unnest(arr) IS NULL)            AS michael     , (SELECT bool_and(e IS NULL) FROM unnest(arr) e)    AS bool_and     , NOT EXISTS (SELECT unnest(arr) EXCEPT SELECT null) AS existFROM  (   VALUES     ('{1,2,NULL,3}'::int[])   , ('{1,1,1}')   , ('{2,2,2}')   , ('{NULL,NULL,NULL}')   , ('{}'::int[])   ) t(arr);       arr        | xsimple | simple | array_rem | array_pos | michael | bool_and | exist ------------------+---------+--------+-----------+-----------+---------+----------+------- {1,2,NULL,3}     | f       | f      | f         | f         | f       | f        | f {1,1,1}          | f       | f      | f         | f         | f       | f        | f {2,2,2}          | f       | f      | f         | f         | f       | f        | f {NULL,NULL,NULL} | t       | t      | t         | t         | t       | t        | t {}               | f       | f      | t         | t         | t       |          | t

array_remove() requires Postgres 9.3 or later.
array_positions() requires Postgres 9.5 or later.

chk_michael is from the currently accepted answer by @michael.
The columns are in order of performance of the expression. Fastest first.
My simple checks dominate performance, with array_remove() next. The rest cannot keep up.

The special case empty array ({}) requires attention. Define the expected result and either pick a fitting expression or add an additional check.

db<>fiddle here - with performance test
_{Old sqlfiddle}

How does it work?

The expression 1 = ALL(arr) yields:

TRUE .. if all elements are 1
FALSE .. if any element is <> 1 (any element that IS NOT NULL)
NULL .. if at least one element IS NULL and no element is <> 1

So, if we know a single element that cannot show up (enforced by a CHECK constraint), like -1, we can simplify to:

-1 = ALL(arr) IS NULL

If any number can show up, check for two distinct numbers. The result can only be NULL for both if the array contains nothing but NULL. Voilá.

I think I got the shortest answer, while still preserving 4 = ALL (ARRAY[4,5]::integer[]); construct:

Live test: https://www.db-fiddle.com/f/6DuB1N4FdcvZdxKiHczu5y/1

selecty, true = ALL (select unnest(z) is null)from x

I'm not exactly proud of this but:

=> select not exists (    select 1    from (select all unnest(ARRAY[NULL, NULL, NULL]) is null as x) as dt    where x = 'f'); ?column? ---------- t(1 row)=> select not exists (    select 1    from (select all unnest(ARRAY[NULL, 11, NULL]) is null as x) as dt    where x = 'f'); ?column? ---------- f(1 row)

Yes, there are subqueries galore but maybe you can make it work or simplify it into something that will work.