if statement doesnt need action

I'm making a user login system with php. Obviously to do that I need to have databases and I do using PHP MyAdmin. The connection in my php to mysql didn't work at first so I looked it up online and found this code:

<?php
$mysql_host = 'localhost';
$mysql_user = 'root';
$mysql_pass = '';

$mysql_db = 'a_database';

if(!mysql_connect($mysql_host, $mysql_user, $mysql_pass)||!mysql_select_db($mysql_db)){
   die(mysql_error());
} 
?>

And this would be enough to connect to SQL. So here is my question: how can it be that you don't have to actually connect to the database in order for it to work? Because to my knowledge of php all that is doing is checking if that works, it isn't actually connecting. So is this code correct? Or do you need to add

mysql_connect($mysql_host, $mysql_user, $mysql_pass)||!mysql_select_db($mysql_db))

to this code to make it work? Thanks in advance.

(I'm using XAMPP btw)