If $X$ is discrete and $Z,W$ are discrete or continuous, is it always the case that $P(X=x\mid Z=z) \geq P(X=x\mid Z=z,W=w)$ for all $x,w,z$? Suppose $X$ is discrete and $Z,W$ are discrete or continuous, I am wondering if it is always the case (or at least non-trivially) that
$$
P(X=x\mid Z=z) \geq P(X=x\mid Z=z,W=w)
$$
for all $x\in X$, $w\in W$, and $z\in Z$?
It appears to make intuitive sense to me because it appears the probability of $X$ given $Z$ and $W$ should be "subsetting" off the probability of $X$ given $Z$. That is, if there is some chance of $X$ given $Z$, then $X$ given $Z$ and $W$ is subdividing the occurrence of $X$ given $Z$ into many more chunks according to $W$, and thus the probability of a chunk occurring should be less than the whole.
In other words, it seems to hold in the finite-sampling perspective, where the above are empirical proportions. However, I am unable to prove this generally. Is there a general result behind this?

Certainly not true in general (in fact in 'most' cases). For example suppose $W=X$ and $X$ is independent of $Z$. If you take $w=x$ then the inequality becomes $P(X=x)^{2} \geq P(X=x)$ so it is false if $0<P(X=x) <1$.
