Theorem elec

Description: Membership in an equivalence class. Theorem 72 of Suppes p. 82. (Contributed by NM, 23-Jul-1995)

	Ref	Expression
Hypotheses	elec.1	⊢ 𝐴 ∈ V
	elec.2	⊢ 𝐵 ∈ V
Assertion	elec	⊢ ( 𝐴 ∈ [ 𝐵 ] 𝑅 ↔ 𝐵 𝑅 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		elec.1	⊢ 𝐴 ∈ V
2		elec.2	⊢ 𝐵 ∈ V
3		elecg	⊢ ( ( 𝐴 ∈ V ∧ 𝐵 ∈ V ) → ( 𝐴 ∈ [ 𝐵 ] 𝑅 ↔ 𝐵 𝑅 𝐴 ) )
4	1 2 3	mp2an	⊢ ( 𝐴 ∈ [ 𝐵 ] 𝑅 ↔ 𝐵 𝑅 𝐴 )