I'm interested in axioms that prevent the existence of large cardinals. More precisely:

(Informal definition) $\Phi$ is an anti-large-cardinal axiom iff $V \models \Phi \Rightarrow V \not \models \Psi$ for some garden-variety large cardinal axiom $\Psi$.

Probably the most famous are:

**Variety 1** Axioms of constructibility (e.g. $V=L$, $V=L[x]$).

$V=L$ for example implies that there are no measurable cardinals in $V$. Often we don't need the full power of $L$ to get anti-large cardinal features though. For instance:

**Variety 2** $\square$-principles

often prevent the existence of large cardinals via their anti-reflection features. For example $\square_\kappa$ prevents the existence of cardinals that reflect stationary sets to $\alpha < \kappa$ (and hence rule out cardinals like supercompacts).

Trivially, we might look at axioms of the following form:

**Variety 3** $\neg \Phi$ or $\neg Con(\Phi)$ for some large cardinal axiom $\Phi$.

These obviously and boringly prevents the existence of a large cardinal in the model. Another possible candidate (formalisable in $NBG + \Sigma^1_1$ comprehension) is:

**Variety 4** Inner model hypotheses.

For example, the vanilla inner model hypothesis that any parameter-free first-order sentence true in an inner model of an outer model of $V$ is already true in an inner model of $V$. This prevents the existence of inaccessibles in $V$. Other variants rule out cardinals not consistent with $L$.

Final example (if we're being super liberal about what we allow to be revised):

**Variety 5** Choice axioms.

For instance $AC$ prevents the existence of Reinhardt (and super-Reinhardt, Berkeley etc.) cardinals in $V$ (assuming that these are, in fact, consistent with $ZF$, which is pretty non-trivial).

My question:

Are there other kinds of axiom with anti-large-cardinal features? Especially non-trivial ones (i.e. not like Variety 3).