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Theorem relnonrel 39940
 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 6040 . . 3 (Rel 𝐴𝐴 = 𝐴)
2 eqss 3981 . . 3 (𝐴 = 𝐴 ↔ (𝐴𝐴𝐴𝐴))
31, 2bitri 277 . 2 (Rel 𝐴 ↔ (𝐴𝐴𝐴𝐴))
4 cnvcnvss 6045 . . 3 𝐴𝐴
54biantrur 533 . 2 (𝐴𝐴 ↔ (𝐴𝐴𝐴𝐴))
6 ssdif0 4322 . 2 (𝐴𝐴 ↔ (𝐴𝐴) = ∅)
73, 5, 63bitr2i 301 1 (Rel 𝐴 ↔ (𝐴𝐴) = ∅)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 208   ∧ wa 398   = wceq 1533   ∖ cdif 3932   ⊆ wss 3935  ∅c0 4290  ◡ccnv 5548  Rel wrel 5554 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 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5195  ax-nul 5202  ax-pr 5321 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-rab 3147  df-v 3496  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4561  df-pr 4563  df-op 4567  df-br 5059  df-opab 5121  df-xp 5555  df-rel 5556  df-cnv 5557 This theorem is referenced by:  cnvnonrel  39941
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