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	$$⊢{\left(A\setminus {A^{-1}}^{-1}\right)}^{-1}=\varnothing$$

Proof

Step	Hyp	Ref	Expression
1		cnvdif	$$⊢{\left(A\setminus {A^{-1}}^{-1}\right)}^{-1}=A^{-1}\setminus {{A^{-1}}^{-1}}^{-1}$$
2		relcnv	$$⊢\mathrm{Rel}{A}^{-1}$$
3		relnonrel	$$⊢\mathrm{Rel}{A}^{-1}\leftrightarrow A^{-1}\setminus {{A^{-1}}^{-1}}^{-1}=\varnothing$$
4	2 3	mpbi	$$⊢A^{-1}\setminus {{A^{-1}}^{-1}}^{-1}=\varnothing$$
5	1 4	eqtri	$$⊢{\left(A\setminus {A^{-1}}^{-1}\right)}^{-1}=\varnothing$$