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Theorem detlem 38164
Description: If a relation is disjoint, then it is equivalent to the equivalent cosets of the relation, inference version. (Contributed by Peter Mazsa, 30-Sep-2021.)
Hypothesis
Ref Expression
detlem.1 Disj 𝑅
Assertion
Ref Expression
detlem ( Disj 𝑅 ↔ EqvRel ≀ 𝑅)

Proof of Theorem detlem
StepHypRef Expression
1 disjim 38162 . 2 ( Disj 𝑅 → EqvRel ≀ 𝑅)
2 detlem.1 . . 3 Disj 𝑅
32a1i 11 . 2 ( EqvRel ≀ 𝑅 → Disj 𝑅)
41, 3impbii 208 1 ( Disj 𝑅 ↔ EqvRel ≀ 𝑅)
Colors of variables: wff setvar class
Syntax hints:  wb 205  ccoss 37554   EqvRel weqvrel 37571   Disj wdisjALTV 37588
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 2163  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-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ral 3056  df-rex 3065  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-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-coss 37792  df-refrel 37893  df-cnvrefrel 37908  df-symrel 37925  df-trrel 37955  df-eqvrel 37966  df-disjALTV 38086
This theorem is referenced by:  det0  38168  detid  38174  detidres  38176  detinidres  38177  detxrnidres  38178
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