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Theorem relnonrel 42914
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 6182 . . 3 (Rel 𝐴𝐴 = 𝐴)
2 eqss 3992 . . 3 (𝐴 = 𝐴 ↔ (𝐴𝐴𝐴𝐴))
31, 2bitri 275 . 2 (Rel 𝐴 ↔ (𝐴𝐴𝐴𝐴))
4 cnvcnvss 6187 . . 3 𝐴𝐴
54biantrur 530 . 2 (𝐴𝐴 ↔ (𝐴𝐴𝐴𝐴))
6 ssdif0 4358 . 2 (𝐴𝐴 ↔ (𝐴𝐴) = ∅)
73, 5, 63bitr2i 299 1 (Rel 𝐴 ↔ (𝐴𝐴) = ∅)
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 395   = wceq 1533  cdif 3940  wss 3943  c0 4317  ccnv 5668  Rel wrel 5674
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-br 5142  df-opab 5204  df-xp 5675  df-rel 5676  df-cnv 5677
This theorem is referenced by:  cnvnonrel  42915
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