Theorem brcnv

Description: The converse of a binary relation swaps arguments. Theorem 11 of Suppes p. 61. (Contributed by NM, 13-Aug-1995)

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

Proof

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