Theorem brvdif2 38222

Description: Binary relation with universal complement. (Contributed by Peter Mazsa, 14-Jul-2018.)

Assertion

Ref	Expression
brvdif2	⊢ (𝐴(V ∖ 𝑅)𝐵 ↔ ¬ ⟨𝐴, 𝐵⟩ ∈ 𝑅)

Proof of Theorem brvdif2

Step	Hyp	Ref	Expression
1		brvdif 38221	. 2 ⊢ (𝐴(V ∖ 𝑅)𝐵 ↔ ¬ 𝐴𝑅𝐵)
2		df-br 5124	. 2 ⊢ (𝐴𝑅𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝑅)
3	1, 2	xchbinx 334	1 ⊢ (𝐴(V ∖ 𝑅)𝐵 ↔ ¬ ⟨𝐴, 𝐵⟩ ∈ 𝑅)

Syntax hints:  ¬ wn 3   ↔ wb 206   ∈ wcel 2107  Vcvv 3463   ∖ cdif 3928  ⟨cop 4612   class class class wbr 5123

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2706  ax-sep 5276  ax-nul 5286  ax-pr 5412

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2713  df-cleq 2726  df-clel 2808  df-v 3465  df-dif 3934  df-un 3936  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-br 5124

This theorem is referenced by: (None)