Theorem br0 5135

Description: The empty binary relation never holds. (Contributed by NM, 23-Aug-2018.)

Assertion

Ref	Expression
br0	⊢ ¬ 𝐴∅𝐵

Proof of Theorem br0

Step	Hyp	Ref	Expression
1		noel 4279	. 2 ⊢ ¬ ⟨𝐴, 𝐵⟩ ∈ ∅
2		df-br 5087	. 2 ⊢ (𝐴∅𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ ∅)
3	1, 2	mtbir 323	1 ⊢ ¬ 𝐴∅𝐵

Syntax hints:  ¬ wn 3   ∈ wcel 2114  ∅c0 4274  ⟨cop 4574   class class class wbr 5086

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

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-dif 3893  df-nul 4275  df-br 5087

This theorem is referenced by:  sbcbr123  5140  sbcbr  5141  cnv0  6097  cnv0OLD  6098  co02  6219  brfvopab  7417  0we1  8434  brdom3  10441  canthwe  10565  relexpindlem  15016  join0  18360  meet0  18361  acycgr0v  35346  prclisacycgr  35349  disjALTV0  39189  brnonrel  44034  upwlkbprop  48626