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Metamath Proof Explorer

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	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → ( 𝐴 ( ( V × V ) ∖ 𝑅 ) 𝐵 ↔ 𝐴 ( V ∖ 𝑅 ) 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		brvvdif	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → ( 𝐴 ( ( V × V ) ∖ 𝑅 ) 𝐵 ↔ ¬ 𝐴 𝑅 𝐵 ) )
2		brvdif	⊢ ( 𝐴 ( V ∖ 𝑅 ) 𝐵 ↔ ¬ 𝐴 𝑅 𝐵 )
3	1 2	bitr4di	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → ( 𝐴 ( ( V × V ) ∖ 𝑅 ) 𝐵 ↔ 𝐴 ( V ∖ 𝑅 ) 𝐵 ) )