Theorem elqs

Description: Membership in a quotient set. (Contributed by NM, 23-Jul-1995)

	Ref	Expression
Hypothesis	elqs.1	$$⊢B\in \mathrm{V}$$
Assertion	elqs	$$⊢B\in \left(A/R\right)\leftrightarrow \exists x\in AB=\left[x\right]R$$

Proof

Step	Hyp	Ref	Expression
1		elqs.1	$$⊢B\in \mathrm{V}$$
2		elqsg	$$⊢B\in \mathrm{V}\to \left(B\in \left(A/R\right)\leftrightarrow \exists x\in AB=\left[x\right]R\right)$$
3	1 2	ax-mp	$$⊢B\in \left(A/R\right)\leftrightarrow \exists x\in AB=\left[x\right]R$$