This principle almost certainly has a name, it has occurred to me that it can be very useful in many areas of math, and perhaps even in life in general, so I want to know its name to remember it better:
Given two statements/conditions $A, B$ such that $A \impliedby B$. In other words, $B$ is a stronger condition than $A$, and and $A$ is weaker (thus more general) than $B$.
Then it follows that $$\{ x\in X: A(x) \text{ is true} \} \supset \{x \in X: B(x) \text{ is true} \} $$
This is an attempt to generalize the accepted answer to a previous question of mine Is the notion of "affineness" more general than "linearity", or vice versa?. This idea is why there are "more" affine subspaces than linear subspaces, and also why $X \subset Y \implies X^{\perp} \supset Y^{\perp}$. Another example is how there being more open sets in finer topologies implies that there are fewer compact sets.
I like this idea in particular because it demonstrates quite nicely how there is often a duality between the partial order of subsets under inclusion and the partial order of statements under implication. I've used this sort of partial order duality (along with the partial order of the real valued functions) many times in probability theory (e.g. $\mathbb{P}(A \cap B) \le \mathbb{P}(A) \le \mathbb{P}(A \cup B)$), so I am not sure why it has never occurred to me before to wonder what its name is.
This question seems to be very related, although not quite the same: Stronger condition lead to weaker result?
