Theorem brvdif2 38481

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 38480	. 2 ⊢ (𝐴(V ∖ 𝑅)𝐵 ↔ ¬ 𝐴𝑅𝐵)
2		df-br 5100	. 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 2114  Vcvv 3441   ∖ cdif 3899  ⟨cop 4587   class class class wbr 5099

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 2709  ax-sep 5242  ax-nul 5252  ax-pr 5378

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-v 3443  df-dif 3905  df-un 3907  df-ss 3919  df-nul 4287  df-if 4481  df-sn 4582  df-pr 4584  df-op 4588  df-br 5100

This theorem is referenced by: (None)