Theorem elqs

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

	Ref	Expression
Hypothesis	elqs.1	⊢ 𝐵 ∈ V
Assertion	elqs	⊢ ( 𝐵 ∈ ( 𝐴 / 𝑅 ) ↔ ∃ 𝑥 ∈ 𝐴 𝐵 = [ 𝑥 ] 𝑅 )

Proof

Step	Hyp	Ref	Expression
1		elqs.1	⊢ 𝐵 ∈ V
2		elqsg	⊢ ( 𝐵 ∈ V → ( 𝐵 ∈ ( 𝐴 / 𝑅 ) ↔ ∃ 𝑥 ∈ 𝐴 𝐵 = [ 𝑥 ] 𝑅 ) )
3	1 2	ax-mp	⊢ ( 𝐵 ∈ ( 𝐴 / 𝑅 ) ↔ ∃ 𝑥 ∈ 𝐴 𝐵 = [ 𝑥 ] 𝑅 )