Oracle 18c

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CREATE TABLE table_name (
  my_json_column CHECK ( my_json_column IS JSON )
) AS
SELECT '{"path":[5,2,1,4,6,19],"not_this_path":[1,2,3,4,5]}' FROM DUAL UNION ALL
SELECT '{"path":[5,2,4,6,19],"not_this_path":[1,2,3,4,5]}' FROM DUAL UNION ALL
SELECT '{"path":[11],"not_this_path":[1]}' FROM DUAL UNION ALL
SELECT '{"path":[2],"not_this_path":[1]}' FROM DUAL UNION ALL
SELECT '{"path":[1,11]}' FROM DUAL;
5 rows affected

SELECT my_json_column,
       CASE
       WHEN EXISTS(
              SELECT 'X'
              FROM   JSON_TABLE(
                       t.my_json_column,
                       '$.path[*]'
                       COLUMNS (
                         value NUMBER PATH '$'
                       )
                     )
              WHERE  value = 1
           )
       THEN 'DIGIT ONE'
       ELSE 'NO DIGIT ONE'
       END AS has_one
FROM   table_name t;


MY_JSON_COLUMN
HAS_ONE


{"path":[5,2,1,4,6,19],"not_this_path":[1,2,3,4,5]}
DIGIT ONE

{"path":[5,2,4,6,19],"not_this_path":[1,2,3,4,5]}
NO DIGIT ONE

{"path":[11],"not_this_path":[1]}
NO DIGIT ONE

{"path":[2],"not_this_path":[1]}
NO DIGIT ONE

{"path":[1,11]}
DIGIT ONE


SELECT my_json_column,
       CASE
       WHEN  JSON_EXISTS( my_json_column, '$?(@.path[*] == 1)' )
       THEN 'DIGIT ONE'
       ELSE 'NO DIGIT ONE'
       END AS has_one
FROM   table_name t;


MY_JSON_COLUMN
HAS_ONE


{"path":[5,2,1,4,6,19],"not_this_path":[1,2,3,4,5]}
DIGIT ONE

{"path":[5,2,4,6,19],"not_this_path":[1,2,3,4,5]}
NO DIGIT ONE

{"path":[11],"not_this_path":[1]}
NO DIGIT ONE

{"path":[2],"not_this_path":[1]}
NO DIGIT ONE

{"path":[1,11]}
DIGIT ONE

MY_JSON_COLUMN	HAS_ONE
{"path":[5,2,1,4,6,19],"not_this_path":[1,2,3,4,5]}	DIGIT ONE
{"path":[5,2,4,6,19],"not_this_path":[1,2,3,4,5]}	NO DIGIT ONE
{"path":[11],"not_this_path":[1]}	NO DIGIT ONE
{"path":[2],"not_this_path":[1]}	NO DIGIT ONE
{"path":[1,11]}	DIGIT ONE