Jamieson, Fausset and Brown on 1 John 5:13

13. The oldest manuscripts and versions read, "These things have I written unto you [omitting 'that believe on the name of the Son of God'] that ye may know that ye have eternal life (compare 1Jo 5:11), THOSE (of you I mean) WHO believe (not as English Version reads, 'and that ye may believe') on the name of the Son of God." English Version, in the latter clause, will mean, "that ye may continue to believe," &c. (compare 1Jo 5:12).

These things--This Epistle. He, towards the close of his Gospel (Joh 20:30, 31), wrote similarly, stating his purpose in having written. In 1Jo 1:4 he states the object of his writing this Epistle to be, "that your joy may be full." To "know that we have eternal life" is the sure way to "joy in God."