Theorem exdifsn

Description: There exists an element in a class excluding a singleton if and only if there exists an element in the original class not equal to the singleton element. (Contributed by BTernaryTau, 15-Sep-2023)

	Ref	Expression
Assertion	exdifsn	$$⊢\exists xx\in \left(A\setminus \left\{B\right\}\right)\leftrightarrow \exists x\in Ax\ne B$$

Proof

Step	Hyp	Ref	Expression
1		eldifsn	$$⊢x\in \left(A\setminus \left\{B\right\}\right)\leftrightarrow \left(x\in A\wedge x\ne B\right)$$
2	1	exbii	$$⊢\exists xx\in \left(A\setminus \left\{B\right\}\right)\leftrightarrow \exists x\left(x\in A\wedge x\ne B\right)$$
3		df-rex	$$⊢\exists x\in Ax\ne B\leftrightarrow \exists x\left(x\in A\wedge x\ne B\right)$$
4	2 3	bitr4i	$$⊢\exists xx\in \left(A\setminus \left\{B\right\}\right)\leftrightarrow \exists x\in Ax\ne B$$