Theorem elqsi

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

	Ref	Expression
Assertion	elqsi	⊢ ( 𝐵 ∈ ( 𝐴 / 𝑅 ) → ∃ 𝑥 ∈ 𝐴 𝐵 = [ 𝑥 ] 𝑅 )

Proof

Step	Hyp	Ref	Expression
1		elqsg	⊢ ( 𝐵 ∈ ( 𝐴 / 𝑅 ) → ( 𝐵 ∈ ( 𝐴 / 𝑅 ) ↔ ∃ 𝑥 ∈ 𝐴 𝐵 = [ 𝑥 ] 𝑅 ) )
2	1	ibi	⊢ ( 𝐵 ∈ ( 𝐴 / 𝑅 ) → ∃ 𝑥 ∈ 𝐴 𝐵 = [ 𝑥 ] 𝑅 )