Theorem cnvnonrel

Description: The converse of the non-relation part of a class is empty. (Contributed by RP, 18-Oct-2020)

	Ref	Expression
Assertion	cnvnonrel	⊢ ◡ ( 𝐴 ∖ ◡ ◡ 𝐴 ) = ∅

Proof

Step	Hyp	Ref	Expression
1		cnvdif	⊢ ◡ ( 𝐴 ∖ ◡ ◡ 𝐴 ) = ( ◡ 𝐴 ∖ ◡ ◡ ◡ 𝐴 )
2		relcnv	⊢ Rel ◡ 𝐴
3		relnonrel	⊢ ( Rel ◡ 𝐴 ↔ ( ◡ 𝐴 ∖ ◡ ◡ ◡ 𝐴 ) = ∅ )
4	2 3	mpbi	⊢ ( ◡ 𝐴 ∖ ◡ ◡ ◡ 𝐴 ) = ∅
5	1 4	eqtri	⊢ ◡ ( 𝐴 ∖ ◡ ◡ 𝐴 ) = ∅