Theorem nonrel

Description: A non-relation is equal to the base class with all ordered pairs removed. (Contributed by RP, 25-Oct-2020)

	Ref	Expression
Assertion	nonrel	$$⊢A\setminus {A^{-1}}^{-1}=A\setminus \left(\mathrm{V}\times \mathrm{V}\right)$$

Proof

Step	Hyp	Ref	Expression
1		cnvcnv	$$⊢{A^{-1}}^{-1}=A\cap \left(\mathrm{V}\times \mathrm{V}\right)$$
2	1	difeq2i	$$⊢A\setminus {A^{-1}}^{-1}=A\setminus \left(A\cap \left(\mathrm{V}\times \mathrm{V}\right)\right)$$
3		difin	$$⊢A\setminus \left(A\cap \left(\mathrm{V}\times \mathrm{V}\right)\right)=A\setminus \left(\mathrm{V}\times \mathrm{V}\right)$$
4	2 3	eqtri	$$⊢A\setminus {A^{-1}}^{-1}=A\setminus \left(\mathrm{V}\times \mathrm{V}\right)$$