Theorem br0 5154

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 4284	. 2 ⊢ ¬ ⟨𝐴, 𝐵⟩ ∈ ∅
2		df-br 5104	. 2 ⊢ (𝐴∅𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ ∅)
3	1, 2	mtbir 326	1 ⊢ ¬ 𝐴∅𝐵

This proof depends on syntax axioms:  ¬ wn 3   ∈ wcel 2145  ∅c0 4279  ⟨cop 4590   class class class wbr 5103

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-dif 3902  df-nul 4280  df-br 5104

This theorem is used by:  sbcbr123  5159  sbcbr  5160  cnv0  5863  cnv0OLD  5864  co02  6257  brfvopab  7470  0we1  8493  brdom3  10531  canthwe  10660  relexpindlem  15136  join0  18491  meet0  18492  acycgr0v  35727  prclisacycgr  35730  disjALTV0  39602  brnonrel  44429  upwlkbprop  49054