>LESS-THAN BUT NOT EQUAL TO

what's the point of this? This is already implied with just "<"

"<" is used to represent partial-order relations, and partial orders are often non-strict (i.e. really less-than-or-equal). Then LESS-THAN BUT NOT EQUAL TO can be used to denote that a particular pair is partially-ordered but not equal (i.e. strict partially ordered). In the same vein there’s also SUBSET OF WITH NOT EQUAL TO (U+228A).

I don't understand. "equal" and "less than" are disjoint. "subset" and "equals" are not disjoint (equal sets are each others subsets), so that's a whole different example.

In mathematics, the symbol "<" is also used for non-strict partial orders, where it effectively means “less than or equal to” instead of “strictly less than”. Working with non-strict partial orders is often easier than working with the equivalent strict partial order, and for economy of notation/ease of writing some mathematicians still use “<“ then instead of “≤”. In the rare cases they want to express strictly-less-than for a particular pair in the non-strict partial order, they can use the “less than but not equals” sign.

> "subset" and "equals" are not disjoint (equal sets are each others subsets)

There is “⊂” vs. “⊆” just like there’s “<“ vs. “≤”. A set is not a strict subset of itself, merely a non-strict subset of itself.

> and for economy of notation/ease of writing some mathematicians still use “<“ then instead of “≤”. In the rare cases they want to express strictly-less-than for a particular pair in the non-strict partial order, they can use the “less than but not equals” sign.

This is pure bullshit of the highest grade. Is is a perfect example of newspeak. "The symbols I use mean what I say they mean whenever I say so". It is gatekeeping. It is part of what makes math impenetrable for a lot of people. It is lazy. It is absurd. Such a lack of consistency is absurd for a rigorous field like math. It is a perfect example of my belief that mathematicians are abysmal linguists. I do not understand why mathematicians have this attitude towards their communication medium. If I could ask a genie for a wish, I would force the use of a compiler and a linter on all mathematicians. If your paper does not pass compilation and linting it gets automatically deleted. Bullshit!

/rant

Thanks, now I understand.

How is the ⪒ used?

I could see people using this symbol when proving for example that x = 0. Say they first show x \geq 0 and then x \leq 0. Then they've shown x ⪒ 0, and therefore x = 0. But yeah, a symbol like '⪒' for this purpose is maybe natural to write on a blackboard, maybe not so much in electronic form.

I'm not aware of any other purposes.

If a ≤ b or b ≤ a. That does not always have to be the case. For example, x ≤ y could mean that x is a descendant of y in a tree.

Interestingly, that last one seems to be missing from LaTeX.