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Theorem disjim 38163
Description: The "Divide et Aequivalere" Theorem: every disjoint relation generates equivalent cosets by the relation: generalization of the former prter1 38261, cf. eldisjim 38166. (Contributed by Peter Mazsa, 3-May-2019.) (Revised by Peter Mazsa, 17-Sep-2021.)
Assertion
Ref Expression
disjim ( Disj 𝑅 → EqvRel ≀ 𝑅)

Proof of Theorem disjim
Dummy variables 𝑢 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dfdisjALTV4 38098 . . . 4 ( Disj 𝑅 ↔ (∀𝑦∃*𝑢 𝑢𝑅𝑦 ∧ Rel 𝑅))
21simplbi 497 . . 3 ( Disj 𝑅 → ∀𝑦∃*𝑢 𝑢𝑅𝑦)
3 trcoss 37864 . . 3 (∀𝑦∃*𝑢 𝑢𝑅𝑦 → ∀𝑥𝑦𝑧((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))
42, 3syl 17 . 2 ( Disj 𝑅 → ∀𝑥𝑦𝑧((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))
5 eqvrelcoss3 38000 . 2 ( EqvRel ≀ 𝑅 ↔ ∀𝑥𝑦𝑧((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))
64, 5sylibr 233 1 ( Disj 𝑅 → EqvRel ≀ 𝑅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wal 1531  ∃*wmo 2526   class class class wbr 5141  Rel wrel 5674  ccoss 37555   EqvRel weqvrel 37572   Disj wdisjALTV 37589
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 37793  df-refrel 37894  df-cnvrefrel 37909  df-symrel 37926  df-trrel 37956  df-eqvrel 37967  df-disjALTV 38087
This theorem is referenced by:  disjimi  38164  detlem  38165  eldisjim  38166  eldisjim2  38167  partim2  38189
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