Theorem br0 5198

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 4331	. 2 ⊢ ¬ ⟨𝐴, 𝐵⟩ ∈ ∅
2		df-br 5150	. 2 ⊢ (𝐴∅𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ ∅)
3	1, 2	mtbir 322	1 ⊢ ¬ 𝐴∅𝐵

Syntax hints:  ¬ wn 3   ∈ wcel 2104  ∅c0 4323  ⟨cop 4635   class class class wbr 5149

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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2701

This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1542  df-fal 1552  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-dif 3952  df-nul 4324  df-br 5150

This theorem is referenced by:  sbcbr123  5203  sbcbr  5204  cnv0  6141  co02  6260  fvmptopabOLD  7468  brfvopab  7470  0we1  8510  brdom3  10527  canthwe  10650  relexpindlem  15016  join0  18364  meet0  18365  acycgr0v  34435  prclisacycgr  34438  disjALTV0  37929  brnonrel  42644  upwlkbprop  46816