Theorem eldifvsn

Description: A set is an element of the universal class excluding a singleton iff it is not the singleton element. (Contributed by AV, 7-Apr-2019)

	Ref	Expression
Assertion	eldifvsn	$$⊢A\in V\to \left(A\in \left(\mathrm{V}\setminus \left\{B\right\}\right)\leftrightarrow A\ne B\right)$$

Proof

Step	Hyp	Ref	Expression
1		eldifsn	$$⊢A\in \left(\mathrm{V}\setminus \left\{B\right\}\right)\leftrightarrow \left(A\in \mathrm{V}\wedge A\ne B\right)$$
2		elex	$$⊢A\in V\to A\in \mathrm{V}$$
3	2	biantrurd	$$⊢A\in V\to \left(A\ne B\leftrightarrow \left(A\in \mathrm{V}\wedge A\ne B\right)\right)$$
4	1 3	bitr4id	$$⊢A\in V\to \left(A\in \left(\mathrm{V}\setminus \left\{B\right\}\right)\leftrightarrow A\ne B\right)$$