Theorem br0 5192

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 4338	. 2 ⊢ ¬ ⟨𝐴, 𝐵⟩ ∈ ∅
2		df-br 5144	. 2 ⊢ (𝐴∅𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ ∅)
3	1, 2	mtbir 323	1 ⊢ ¬ 𝐴∅𝐵

Syntax hints:  ¬ wn 3   ∈ wcel 2108  ∅c0 4333  ⟨cop 4632   class class class wbr 5143

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-dif 3954  df-nul 4334  df-br 5144

This theorem is referenced by:  sbcbr123  5197  sbcbr  5198  cnv0  6160  co02  6280  fvmptopabOLD  7488  brfvopab  7490  0we1  8544  brdom3  10568  canthwe  10691  relexpindlem  15102  join0  18450  meet0  18451  acycgr0v  35153  prclisacycgr  35156  disjALTV0  38755  brnonrel  43602  upwlkbprop  48054