Theorem brvdif 38726

Description: Binary relation with universal complement is the negation of the relation. (Contributed by Peter Mazsa, 1-Jul-2018.)

Assertion

Ref	Expression
brvdif	⊢ (𝐴(V ∖ 𝑅)𝐵 ↔ ¬ 𝐴𝑅𝐵)

Proof of Theorem brvdif

Step	Hyp	Ref	Expression
1		brv 5437	. 2 ⊢ 𝐴V𝐵
2		brdif 5150	. 2 ⊢ (𝐴(V ∖ 𝑅)𝐵 ↔ (𝐴V𝐵 ∧ ¬ 𝐴𝑅𝐵))
3	1, 2	mpbiran 719	1 ⊢ (𝐴(V ∖ 𝑅)𝐵 ↔ ¬ 𝐴𝑅𝐵)

Syntax hints:  ¬ wn 3   ↔ wb 208  Vcvv 3453   ∖ cdif 3899   class class class wbr 5097

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733  ax-sep 5243  ax-pr 5387

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-rab 3414  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-sn 4580  df-pr 4582  df-op 4586  df-br 5098

This theorem is referenced by:  brvdif2  38727  brvbrvvdif  38729  dfssr2  39039