Theorem ercl2

Description: Elementhood in the field of an equivalence relation. (Contributed by Mario Carneiro, 12-Aug-2015)

	Ref	Expression
Hypotheses	ersym.1	⊢ ( 𝜑 → 𝑅 Er 𝑋 )
	ersym.2	⊢ ( 𝜑 → 𝐴 𝑅 𝐵 )
Assertion	ercl2	⊢ ( 𝜑 → 𝐵 ∈ 𝑋 )

Proof

Step	Hyp	Ref	Expression
1		ersym.1	⊢ ( 𝜑 → 𝑅 Er 𝑋 )
2		ersym.2	⊢ ( 𝜑 → 𝐴 𝑅 𝐵 )
3	1 2	ersym	⊢ ( 𝜑 → 𝐵 𝑅 𝐴 )
4	1 3	ercl	⊢ ( 𝜑 → 𝐵 ∈ 𝑋 )