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Theorem br0 4631
 Description: The empty binary relation never holds. (Contributed by NM, 23-Aug-2018.)
Assertion
Ref Expression
br0 ¬ 𝐴𝐵

Proof of Theorem br0
StepHypRef Expression
1 noel 3878 . 2 ¬ ⟨𝐴, 𝐵⟩ ∈ ∅
2 df-br 4584 . 2 (𝐴𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ ∅)
31, 2mtbir 312 1 ¬ 𝐴𝐵
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ∈ wcel 1977  ∅c0 3874  ⟨cop 4131   class class class wbr 4583 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175  df-dif 3543  df-nul 3875  df-br 4584 This theorem is referenced by:  sbcbr123  4636  sbcbr  4637  cnv0  5454  co02  5566  brfvopab  6598  0we1  7473  brdom3  9231  canthwe  9352  meet0  16960  join0  16961  brnonrel  36914  wlkbProp  40817
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