Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  br0 GIF version

Theorem br0 3971
 Description: The empty binary relation never holds. (Contributed by NM, 23-Aug-2018.)
Assertion
Ref Expression
br0 ¬ 𝐴𝐵

Proof of Theorem br0
StepHypRef Expression
1 noel 3362 . 2 ¬ ⟨𝐴, 𝐵⟩ ∈ ∅
2 df-br 3925 . 2 (𝐴𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ ∅)
31, 2mtbir 660 1 ¬ 𝐴𝐵
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   ∈ wcel 1480  ∅c0 3358  ⟨cop 3525   class class class wbr 3924 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119 This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-v 2683  df-dif 3068  df-nul 3359  df-br 3925 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator