Notation for set excluding element

Consider the set $\mathbb{A}=\{a,b,c\}$.

I want to refer to "the set $\mathbb{A}$ excluding the element $a$"

Is the notation $\mathbb{A} \sim \{a\}$ equivalent to $\mathbb{A} \setminus \{a\}$?

Is the former an abuse of logic notation?

The latter is what I am used to.

The notation $\Bbb A - \{a\}$ is often used to mean the same thing as $\Bbb A \setminus \{a\}$ (the set difference), but I've never seen it with a tilde and can't find any references to it being used this way with Google.

The tilde $\sim$ is sometimes used as a negation or "not" symbol in set theory, in which case

$$\Bbb A \setminus \{a\} = \bigl\{x : x \in \Bbb A, \sim\!(x\in\{a\})\bigr\}.$$

The tilde is also used sometimes for equivalence relations, where $x \sim y$ means $x$ and $y$ are equivalent (in the same equivalence class) under some equivalence relation $\sim$.

A particularly common example of this is with the cardinality of sets. We say $A \sim B$ if $A$ and $B$ have the same cardinality, that is $|A| = |B|$, and we call them equinumerous.