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Theorem relnonrel 41866
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 6142 . . 3 (Rel 𝐴𝐴 = 𝐴)
2 eqss 3960 . . 3 (𝐴 = 𝐴 ↔ (𝐴𝐴𝐴𝐴))
31, 2bitri 275 . 2 (Rel 𝐴 ↔ (𝐴𝐴𝐴𝐴))
4 cnvcnvss 6147 . . 3 𝐴𝐴
54biantrur 532 . 2 (𝐴𝐴 ↔ (𝐴𝐴𝐴𝐴))
6 ssdif0 4324 . 2 (𝐴𝐴 ↔ (𝐴𝐴) = ∅)
73, 5, 63bitr2i 299 1 (Rel 𝐴 ↔ (𝐴𝐴) = ∅)
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 397   = wceq 1542  cdif 3908  wss 3911  c0 4283  ccnv 5633  Rel wrel 5639
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2708  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-rab 3409  df-v 3448  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-br 5107  df-opab 5169  df-xp 5640  df-rel 5641  df-cnv 5642
This theorem is referenced by:  cnvnonrel  41867
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