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Theorem disjALTVid 38283
Description: The class of identity relations is disjoint. (Contributed by Peter Mazsa, 20-Jun-2021.)
Assertion
Ref Expression
disjALTVid Disj I

Proof of Theorem disjALTVid
StepHypRef Expression
1 cosscnvid 38009 . . 3 I = I
21eqimssi 4033 . 2 I ⊆ I
3 reli 5822 . 2 Rel I
4 dfdisjALTV2 38242 . 2 ( Disj I ↔ ( ≀ I ⊆ I ∧ Rel I ))
52, 3, 4mpbir2an 709 1 Disj I
Colors of variables: wff setvar class
Syntax hints:  wss 3939   I cid 5569  ccnv 5671  Rel wrel 5677  ccoss 37705   Disj wdisjALTV 37739
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-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-br 5144  df-opab 5206  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-coss 37939  df-cnvrefrel 38055  df-disjALTV 38233
This theorem is referenced by:  disjALTVidres  38284  disjALTVinidres  38285  disjALTVxrnidres  38286  eqvrelid  38317  detid  38321  eqvrelcossid  38322  petid2  38344
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