Theorem brvbrvvdif

Description: Binary relation with the complement under the universal class of ordered pairs is the same as with universal complement. (Contributed by Peter Mazsa, 28-Nov-2018)

	Ref	Expression
Assertion	brvbrvvdif	$$⊢\left(A\in V\wedge B\in W\right)\to \left(A\left(\left(\mathrm{V}\times \mathrm{V}\right)\setminus R\right)B\leftrightarrow A\left(\mathrm{V}\setminus R\right)B\right)$$

Proof

Step	Hyp	Ref	Expression
1		brvvdif	$$⊢\left(A\in V\wedge B\in W\right)\to \left(A\left(\left(\mathrm{V}\times \mathrm{V}\right)\setminus R\right)B\leftrightarrow \neg ARB\right)$$
2		brvdif	$$⊢A\left(\mathrm{V}\setminus R\right)B\leftrightarrow \neg ARB$$
3	1 2	bitr4di	$$⊢\left(A\in V\wedge B\in W\right)\to \left(A\left(\left(\mathrm{V}\times \mathrm{V}\right)\setminus R\right)B\leftrightarrow A\left(\mathrm{V}\setminus R\right)B\right)$$