Theorem brvdif 39014

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 5448	. 2 ⊢ 𝐴V𝐵
2		brdif 5158	. 2 ⊢ (𝐴(V ∖ 𝑅)𝐵 ↔ (𝐴V𝐵 ∧ ¬ 𝐴𝑅𝐵))
3	1, 2	mpbiran 722	1 ⊢ (𝐴(V ∖ 𝑅)𝐵 ↔ ¬ 𝐴𝑅𝐵)

This proof depends on syntax axioms:  ¬ wn 3   ↔ wb 209  Vcvv 3450   ∖ cdif 3896   class class class wbr 5103

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732  ax-sep 5251  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-rab 3413  df-v 3452  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-sn 4585  df-pr 4587  df-op 4591  df-br 5104

This theorem is used by:  brvdif2  39015  brvbrvvdif  39017  dfssr2  39327