Theorem eueq

Description: A class is a set if and only if there exists a unique set equal to it. (Contributed by NM, 25-Nov-1994) Shorten combined proofs of moeq and eueq . (Proof shortened by BJ, 24-Sep-2022)

	Ref	Expression
Assertion	eueq	$$⊢A\in \mathrm{V}\leftrightarrow \exists !xx=A$$

Proof

Step	Hyp	Ref	Expression
1		moeq	$$⊢{\exists }^{*}xx=A$$
2	1	biantru	$$⊢\exists xx=A\leftrightarrow \left(\exists xx=A\wedge {\exists }^{*}xx=A\right)$$
3		isset	$$⊢A\in \mathrm{V}\leftrightarrow \exists xx=A$$
4		df-eu	$$⊢\exists !xx=A\leftrightarrow \left(\exists xx=A\wedge {\exists }^{*}xx=A\right)$$
5	2 3 4	3bitr4i	$$⊢A\in \mathrm{V}\leftrightarrow \exists !xx=A$$