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Theorem relnonrel 43577
Description: The non-relation part of a relation is empty. (Contributed by RP, 22-Oct-2020.)
Assertion
Ref Expression
relnonrel (Rel 𝐴 ↔ (𝐴𝐴) = ∅)

Proof of Theorem relnonrel
StepHypRef Expression
1 dfrel2 6211 . . 3 (Rel 𝐴𝐴 = 𝐴)
2 eqss 4011 . . 3 (𝐴 = 𝐴 ↔ (𝐴𝐴𝐴𝐴))
31, 2bitri 275 . 2 (Rel 𝐴 ↔ (𝐴𝐴𝐴𝐴))
4 cnvcnvss 6216 . . 3 𝐴𝐴
54biantrur 530 . 2 (𝐴𝐴 ↔ (𝐴𝐴𝐴𝐴))
6 ssdif0 4372 . 2 (𝐴𝐴 ↔ (𝐴𝐴) = ∅)
73, 5, 63bitr2i 299 1 (Rel 𝐴 ↔ (𝐴𝐴) = ∅)
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395   = wceq 1537  cdif 3960  wss 3963  c0 4339  ccnv 5688  Rel wrel 5694
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 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-xp 5695  df-rel 5696  df-cnv 5697
This theorem is referenced by:  cnvnonrel  43578
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