Theorem br0 5160

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 4286	. 2 ⊢ ¬ ⟨𝐴, 𝐵⟩ ∈ ∅
2		df-br 5109	. 2 ⊢ (𝐴∅𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ ∅)
3	1, 2	mtbir 326	1 ⊢ ¬ 𝐴∅𝐵

This proof depends on syntax axioms:  ¬ wn 3   ∈ wcel 2147  ∅c0 4281  ⟨cop 4840   class class class wbr 5108

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-dif 4050  df-nul 4282  df-br 5109

This theorem is used by:  sbcbr123  5166  sbcbr  5167  cnv0  5873  cnv0OLD  5874  co02  6283  brfvopab  7508  0we1  8552  brdom3  10646  canthwe  10775  relexpindlem  15259  join0  18627  meet0  18628  acycgr0v  36111  prclisacycgr  36114  disjALTV0  40052  brnonrel  44850  upwlkbprop  49509