A Mail Transfer Agent will request the MX DNS entries for the receiving domain (example.com) and determine the record with the lowest distance (priority or preference) (mail.example.com, although no preference is given in your example) and will look up the corresponding IP address for that record. The MTA will connect to the IP address returned by the DNS server to deliver the e-mail.
In your example you are missing a A record for mail.example.com and a preference for the MX record, but the notation of the preference depends on the used DNS software.
Name | Type | Value
example.com | A | 22.214.171.124
example.com | MX | 10 mail.example.com
mail.example.com | A | 126.96.36.199