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Theorem brne0 5107
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 5058 . 2 (𝐴𝑅𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝑅)
2 ne0i 4297 . 2 (⟨𝐴, 𝐵⟩ ∈ 𝑅𝑅 ≠ ∅)
31, 2sylbi 218 1 (𝐴𝑅𝐵𝑅 ≠ ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2105  wne 3013  c0 4288  cop 4563   class class class wbr 5057
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1772  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-ne 3014  df-dif 3936  df-nul 4289  df-br 5058
This theorem is referenced by:  epn0  5464  brfvopabrbr  6758  bropfvvvvlem  7775  brfvimex  40254  brovmptimex  40255  clsneibex  40330  neicvgbex  40340
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