Theorem br0 5154

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 4284	. 2 ⊢ ¬ ⟨𝐴, 𝐵⟩ ∈ ∅
2		df-br 5104	. 2 ⊢ (𝐴∅𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ ∅)
3	1, 2	mtbir 326	1 ⊢ ¬ 𝐴∅𝐵

This proof depends on syntax axioms:  ¬ wn 3   ∈ wcel 2145  ∅c0 4279  ⟨cop 4590   class class class wbr 5103

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-dif 3902  df-nul 4280  df-br 5104

This theorem is used by:  sbcbr123  5159  sbcbr  5160  cnv0  5861  cnv0OLD  5862  co02  6261  brfvopab  7475  0we1  8507  brdom3  10600  canthwe  10729  relexpindlem  15209  join0  18570  meet0  18571  acycgr0v  35892  prclisacycgr  35895  disjALTV0  39766  brnonrel  44574  upwlkbprop  49205