Theorem relelec

Description: Membership in an equivalence class when R is a relation. (Contributed by Mario Carneiro, 11-Sep-2015)

	Ref	Expression
Assertion	relelec	⊢ ( Rel 𝑅 → ( 𝐴 ∈ [ 𝐵 ] 𝑅 ↔ 𝐵 𝑅 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		elex	⊢ ( 𝐴 ∈ [ 𝐵 ] 𝑅 → 𝐴 ∈ V )
2		ecexr	⊢ ( 𝐴 ∈ [ 𝐵 ] 𝑅 → 𝐵 ∈ V )
3	1 2	jca	⊢ ( 𝐴 ∈ [ 𝐵 ] 𝑅 → ( 𝐴 ∈ V ∧ 𝐵 ∈ V ) )
4	3	adantl	⊢ ( ( Rel 𝑅 ∧ 𝐴 ∈ [ 𝐵 ] 𝑅 ) → ( 𝐴 ∈ V ∧ 𝐵 ∈ V ) )
5		brrelex12	⊢ ( ( Rel 𝑅 ∧ 𝐵 𝑅 𝐴 ) → ( 𝐵 ∈ V ∧ 𝐴 ∈ V ) )
6	5	ancomd	⊢ ( ( Rel 𝑅 ∧ 𝐵 𝑅 𝐴 ) → ( 𝐴 ∈ V ∧ 𝐵 ∈ V ) )
7		elecg	⊢ ( ( 𝐴 ∈ V ∧ 𝐵 ∈ V ) → ( 𝐴 ∈ [ 𝐵 ] 𝑅 ↔ 𝐵 𝑅 𝐴 ) )
8	4 6 7	pm5.21nd	⊢ ( Rel 𝑅 → ( 𝐴 ∈ [ 𝐵 ] 𝑅 ↔ 𝐵 𝑅 𝐴 ) )