MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  brne0 Structured version   Visualization version   GIF version

Theorem brne0 5109
Description: If two sets are in a binary relation, the relation cannot be empty. (Contributed by Alexander van der Vekens, 7-Jul-2018.)
Assertion
Ref Expression
brne0 (𝐴𝑅𝐵𝑅 ≠ ∅)

Proof of Theorem brne0
StepHypRef Expression
1 df-br 5060 . 2 (𝐴𝑅𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝑅)
2 ne0i 4300 . 2 (⟨𝐴, 𝐵⟩ ∈ 𝑅𝑅 ≠ ∅)
31, 2sylbi 219 1 (𝐴𝑅𝐵𝑅 ≠ ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2110  wne 3016  c0 4291  cop 4567   class class class wbr 5059
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-ext 2793
This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1777  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-ne 3017  df-dif 3939  df-nul 4292  df-br 5060
This theorem is referenced by:  epn0  5466  brfvopabrbr  6760  bropfvvvvlem  7780  brfvimex  40369  brovmptimex  40370  clsneibex  40445  neicvgbex  40455
  Copyright terms: Public domain W3C validator