Theorem br0 5196

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 4329	. 2 ⊢ ¬ ⟨𝐴, 𝐵⟩ ∈ ∅
2		df-br 5148	. 2 ⊢ (𝐴∅𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ ∅)
3	1, 2	mtbir 323	1 ⊢ ¬ 𝐴∅𝐵

Syntax hints:  ¬ wn 3   ∈ wcel 2107  ∅c0 4321  ⟨cop 4633   class class class wbr 5147

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704

This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-dif 3950  df-nul 4322  df-br 5148

This theorem is referenced by:  sbcbr123  5201  sbcbr  5202  cnv0  6137  co02  6256  fvmptopabOLD  7459  brfvopab  7461  0we1  8501  brdom3  10519  canthwe  10642  relexpindlem  15006  join0  18354  meet0  18355  acycgr0v  34077  prclisacycgr  34080  disjALTV0  37562  brnonrel  42273  upwlkbprop  46451