Theorem opelcnv

Description: Ordered-pair membership in converse relation. (Contributed by NM, 13-Aug-1995)

	Ref	Expression
Hypotheses	opelcnv.1	⊢ 𝐴 ∈ V
	opelcnv.2	⊢ 𝐵 ∈ V
Assertion	opelcnv	⊢ ( ⟨ 𝐴 , 𝐵 ⟩ ∈ ◡ 𝑅 ↔ ⟨ 𝐵 , 𝐴 ⟩ ∈ 𝑅 )

Proof

Step	Hyp	Ref	Expression
1		opelcnv.1	⊢ 𝐴 ∈ V
2		opelcnv.2	⊢ 𝐵 ∈ V
3		opelcnvg	⊢ ( ( 𝐴 ∈ V ∧ 𝐵 ∈ V ) → ( ⟨ 𝐴 , 𝐵 ⟩ ∈ ◡ 𝑅 ↔ ⟨ 𝐵 , 𝐴 ⟩ ∈ 𝑅 ) )
4	1 2 3	mp2an	⊢ ( ⟨ 𝐴 , 𝐵 ⟩ ∈ ◡ 𝑅 ↔ ⟨ 𝐵 , 𝐴 ⟩ ∈ 𝑅 )