For every set exists another stronger set

Is the following claim true?

For every set exists another stronger set.

By "if $A$ is stronger than $B$", we mean that $B$ is equinumerous to a subset of $A$, and $A$ is equinumerous to none of the subsets of $B$.

The question had been asked in a class and the professor used Cantor’s theorem to prove the claim.

I think the statement can be wittern formally as:

$$forall A,exists B: text{card}(A) < text{Card}(B)$$

One may use Cantor’s theorem as the professor did, but I think the claim is true as long as the universal set does not exist because if it does and is denoted by $U$ then from the definition we know that every other set is contained in $U$ and so the negation of the claim does hold.

I like to know whether I’m right and it’s interesting to know more about the claim, thanks in advance.