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Theorem br0 4661
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 3895 . 2 ¬ ⟨𝐴, 𝐵⟩ ∈ ∅
2 df-br 4614 . 2 (𝐴𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ ∅)
31, 2mtbir 313 1 ¬ 𝐴𝐵
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wcel 1987  c0 3891  cop 4154   class class class wbr 4613
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-v 3188  df-dif 3558  df-nul 3892  df-br 4614
This theorem is referenced by:  sbcbr123  4666  sbcbr  4667  cnv0  5494  co02  5608  fvmptopab  6650  brfvopab  6653  0we1  7531  brdom3  9294  canthwe  9417  meet0  17058  join0  17059  brnonrel  37376  upwlkbprop  41007
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