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

Theorem brne0 5198
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 5149 . 2 (𝐴𝑅𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝑅)
2 ne0i 4335 . 2 (⟨𝐴, 𝐵⟩ ∈ 𝑅𝑅 ≠ ∅)
31, 2sylbi 216 1 (𝐴𝑅𝐵𝑅 ≠ ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2099  wne 2937  c0 4323  cop 4635   class class class wbr 5148
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699
This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-ne 2938  df-dif 3950  df-nul 4324  df-br 5149
This theorem is referenced by:  epn0  5587  brfvopabrbr  7002  bropfvvvvlem  8096  brfvimex  43456  brovmptimex  43457  clsneibex  43532  neicvgbex  43542
  Copyright terms: Public domain W3C validator