Theorem snec

Description: The singleton of an equivalence class. (Contributed by NM, 29-Jan-1999) (Revised by Mario Carneiro, 9-Jul-2014)

	Ref	Expression
Hypothesis	snec.1	$$⊢A\in \mathrm{V}$$
Assertion	snec	$$⊢\left\{\left[A\right]R\right\}=\left\{A\right\}/R$$

Proof

Step	Hyp	Ref	Expression
1		snec.1	$$⊢A\in \mathrm{V}$$
2		eceq1	$$⊢x=A\to \left[x\right]R=\left[A\right]R$$
3	2	eqeq2d	$$⊢x=A\to \left(y=\left[x\right]R\leftrightarrow y=\left[A\right]R\right)$$
4	1 3	rexsn	$$⊢\exists x\in \left\{A\right\}y=\left[x\right]R\leftrightarrow y=\left[A\right]R$$
5	4	abbii	$$⊢\left\{y|\exists x\in \left\{A\right\}y=\left[x\right]R\right\}=\left\{y|y=\left[A\right]R\right\}$$
6		df-qs	$$⊢\left\{A\right\}/R=\left\{y|\exists x\in \left\{A\right\}y=\left[x\right]R\right\}$$
7		df-sn	$$⊢\left\{\left[A\right]R\right\}=\left\{y|y=\left[A\right]R\right\}$$
8	5 6 7	3eqtr4ri	$$⊢\left\{\left[A\right]R\right\}=\left\{A\right\}/R$$