Theorem br0 5146

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 4288	. 2 ⊢ ¬ ⟨𝐴, 𝐵⟩ ∈ ∅
2		df-br 5098	. 2 ⊢ (𝐴∅𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ ∅)
3	1, 2	mtbir 325	1 ⊢ ¬ 𝐴∅𝐵

Syntax hints:  ¬ wn 3   ∈ wcel 2141  ∅c0 4283  ⟨cop 4585   class class class wbr 5097

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733

This theorem depends on definitions:  df-bi 209  df-an 400  df-tru 1562  df-fal 1572  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-dif 3905  df-nul 4284  df-br 5098

This theorem is referenced by:  sbcbr123  5151  sbcbr  5152  cnv0  5851  cnv0OLD  5852  co02  6243  brfvopab  7448  0we1  8469  brdom3  10479  canthwe  10603  relexpindlem  15070  join0  18426  meet0  18427  acycgr0v  35459  prclisacycgr  35462  disjALTV0  39314  brnonrel  44126  upwlkbprop  48721