Theorem br0 5144

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 4287	. 2 ⊢ ¬ ⟨𝐴, 𝐵⟩ ∈ ∅
2		df-br 5096	. 2 ⊢ (𝐴∅𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ ∅)
3	1, 2	mtbir 323	1 ⊢ ¬ 𝐴∅𝐵

Syntax hints:  ¬ wn 3   ∈ wcel 2113  ∅c0 4282  ⟨cop 4583   class class class wbr 5095

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2705

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2712  df-cleq 2725  df-clel 2808  df-dif 3901  df-nul 4283  df-br 5096

This theorem is referenced by:  sbcbr123  5149  sbcbr  5150  cnv0  6094  cnv0OLD  6095  co02  6216  brfvopab  7412  0we1  8430  brdom3  10430  canthwe  10553  relexpindlem  14977  join0  18317  meet0  18318  acycgr0v  35264  prclisacycgr  35267  disjALTV0  38925  brnonrel  43746  upwlkbprop  48300