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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