Theorem brvdif 38587

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 5425	. 2 ⊢ 𝐴V𝐵
2		brdif 5138	. 2 ⊢ (𝐴(V ∖ 𝑅)𝐵 ↔ (𝐴V𝐵 ∧ ¬ 𝐴𝑅𝐵))
3	1, 2	mpbiran 710	1 ⊢ (𝐴(V ∖ 𝑅)𝐵 ↔ ¬ 𝐴𝑅𝐵)

Syntax hints:  ¬ wn 3   ↔ wb 206  Vcvv 3429   ∖ cdif 3886   class class class wbr 5085

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2708  ax-sep 5231  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-sn 4568  df-pr 4570  df-op 4574  df-br 5086

This theorem is referenced by:  brvdif2  38588  brvbrvvdif  38590  dfssr2  38900