MySQL 8.0

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Create Table TableA(class_id VARCHAR(50), prop_id VARCHAR(50));
Insert Into TableA Values
('12','aa_13'),
('13','ab_21'),
('22','ac_11'),
('53','bb_32'),
('48','ac_57');
Create Table TableB(class_id INT, prop_id VARCHAR(50), date_from DATE, date_to DATE);
Insert Into TableB Values
(12,'aa_13','2022-02-15','2022-12-10'),
(12,'aa_31','2021-09-30','2022-02-12'),
(13,'ac_12','2021-05-18','2022-02-05'),
(22,'ac_11','2022-05-12','2022-08-25'),
(22,'ac_12','2022-01-05','2022-04-23'),
(22,'ac_13','2021-08-18','2021-11-16'),
(53,'bb_32','2022-02-06','2022-03-19'),
(53,'bb_31','2021-05-08','2022-02-05'),
(48,'ac_57','2022-02-03','2022-05-07');
With CTE As
(
  Select Distinct class_id
  From TableB
  Where date_from <= '2022-05-01'
        and date_to >= '2022-05-31'
)

Select A.class_id, 
       IF(C.class_id IS NULL, NULL, A.prop_id) prop_id
From 
TableA A Left Join CTE C
on A.class_id = C.class_id


class_id
prop_id


12
aa_13

13
null

22
null

53
null

48
null

class_id	prop_id
12	aa_13
13	null
22	null
53	null
48	null