Theorem br0 5119

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 4261	. 2 ⊢ ¬ ⟨𝐴, 𝐵⟩ ∈ ∅
2		df-br 5071	. 2 ⊢ (𝐴∅𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ ∅)
3	1, 2	mtbir 322	1 ⊢ ¬ 𝐴∅𝐵

Syntax hints:  ¬ wn 3   ∈ wcel 2108  ∅c0 4253  ⟨cop 4564   class class class wbr 5070

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-dif 3886  df-nul 4254  df-br 5071

This theorem is referenced by:  sbcbr123  5124  sbcbr  5125  cnv0  6033  co02  6153  fvmptopab  7308  brfvopab  7310  0we1  8298  brdom3  10215  canthwe  10338  relexpindlem  14702  join0  18038  meet0  18039  acycgr0v  33010  prclisacycgr  33013  disjALTV0  36789  brnonrel  41086  upwlkbprop  45188