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 323	1 ⊢ ¬ 𝐴∅𝐵

Syntax hints:  ¬ wn 3   ∈ wcel 2107  ∅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 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704

This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-dif 3952  df-nul 4324  df-br 5150

This theorem is referenced by:  sbcbr123  5203  sbcbr  5204  cnv0  6141  co02  6260  fvmptopabOLD  7464  brfvopab  7466  0we1  8506  brdom3  10523  canthwe  10646  relexpindlem  15010  join0  18358  meet0  18359  acycgr0v  34139  prclisacycgr  34142  disjALTV0  37624  brnonrel  42340  upwlkbprop  46516