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Theorem brne0 4891
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 4842 . 2 (𝐴𝑅𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝑅)
2 ne0i 4119 . 2 (⟨𝐴, 𝐵⟩ ∈ 𝑅𝑅 ≠ ∅)
31, 2sylbi 209 1 (𝐴𝑅𝐵𝑅 ≠ ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2157  wne 2969  c0 4113  cop 4372   class class class wbr 4841
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-ext 2775
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2784  df-cleq 2790  df-clel 2793  df-nfc 2928  df-ne 2970  df-v 3385  df-dif 3770  df-nul 4114  df-br 4842
This theorem is referenced by:  epn0  5228  brfvopabrbr  6502  bropfvvvvlem  7491  brfvimex  39094  brovmptimex  39095  clsneibex  39170  neicvgbex  39180
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