Theorem br0 5149

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 4292	. 2 ⊢ ¬ ⟨𝐴, 𝐵⟩ ∈ ∅
2		df-br 5101	. 2 ⊢ (𝐴∅𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ ∅)
3	1, 2	mtbir 323	1 ⊢ ¬ 𝐴∅𝐵

Syntax hints:  ¬ wn 3   ∈ wcel 2114  ∅c0 4287  ⟨cop 4588   class class class wbr 5100

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 3906  df-nul 4288  df-br 5101

This theorem is referenced by:  sbcbr123  5154  sbcbr  5155  cnv0  6105  cnv0OLD  6106  co02  6227  brfvopab  7425  0we1  8443  brdom3  10450  canthwe  10574  relexpindlem  14998  join0  18338  meet0  18339  acycgr0v  35361  prclisacycgr  35364  disjALTV0  39102  brnonrel  43942  upwlkbprop  48495