Why “ every non-prime attribute of R is non-transitively dependent on every key of R” in the original def of 3NF?

Question

I have read the definition in of Third_normal_form here , which shows

The third normal form (3NF) is a normal form used in database normalization. 3NF was originally defined by E.F. Codd in 1971.[2] Codd's definition states that a table is in 3NF if and only if both of the following conditions hold:

1. The relation R (table) is in second normal form (2NF)
2. Every non-prime attribute of R is non-transitively dependent on every key of R.

From the book Database Systems The Complete Book 2nd Edition, key means candidate key.

And I got the definition of transitive dependent here. So my problem is this: From the quoted definition, if we denote any non-prime attr as $x$, any key of $R$ as $K$, suppose $x$ is subset of an attributes group $A$, we can easily construct some $A$ satisfied $$K \rightarrow A \wedge A \nrightarrow K \wedge A\rightarrow x $$ so where am I wrong?