[mathgradthrow]

No. Equality is defined on sets, with exactly one exception, this stupid fucking notation.

[foxes]

Notation is invented to be useful. A lot of mathematicians and physicists write stuff like

f = g + O(h)

and then proceed to do a bunch of manipulations without getting bogged down in set theory syntax.

Also equality isnt just defined on sets, usually people start by defining it as like a formal thing satisfying a bunch of rules as a part of their logic, and going from there.

[mathgradthrow]

Equality is defined on sets. Everything is a set. Equality means equalty as sets. That's how math aorks unless you're doing type theory shit.