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Theorem br0 4809
 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 4027 . 2 ¬ ⟨𝐴, 𝐵⟩ ∈ ∅
2 df-br 4761 . 2 (𝐴𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ ∅)
31, 2mtbir 312 1 ¬ 𝐴𝐵
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ∈ wcel 2103  ∅c0 4023  ⟨cop 4291   class class class wbr 4760 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-v 3306  df-dif 3683  df-nul 4024  df-br 4761 This theorem is referenced by:  sbcbr123  4814  sbcbr  4815  cnv0  5645  co02  5762  fvmptopab  6814  brfvopab  6817  0we1  7706  brdom3  9463  canthwe  9586  meet0  17259  join0  17260  brnonrel  38314  upwlkbprop  42146
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