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Theorem disjim 38739
Description: The "Divide et Aequivalere" Theorem: every disjoint relation generates equivalent cosets by the relation: generalization of the former prter1 38837, cf. eldisjim 38742. (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 38674 . . . 4 ( Disj 𝑅 ↔ (∀𝑦∃*𝑢 𝑢𝑅𝑦 ∧ Rel 𝑅))
21simplbi 497 . . 3 ( Disj 𝑅 → ∀𝑦∃*𝑢 𝑢𝑅𝑦)
3 trcoss 38440 . . 3 (∀𝑦∃*𝑢 𝑢𝑅𝑦 → ∀𝑥𝑦𝑧((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))
42, 3syl 17 . 2 ( Disj 𝑅 → ∀𝑥𝑦𝑧((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))
5 eqvrelcoss3 38576 . 2 ( EqvRel ≀ 𝑅 ↔ ∀𝑥𝑦𝑧((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))
64, 5sylibr 234 1 ( Disj 𝑅 → EqvRel ≀ 𝑅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wal 1535  ∃*wmo 2541   class class class wbr 5166  Rel wrel 5705  ccoss 38137   EqvRel weqvrel 38154   Disj wdisjALTV 38171
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-br 5167  df-opab 5229  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-coss 38369  df-refrel 38470  df-cnvrefrel 38485  df-symrel 38502  df-trrel 38532  df-eqvrel 38543  df-disjALTV 38663
This theorem is referenced by:  disjimi  38740  detlem  38741  eldisjim  38742  eldisjim2  38743  partim2  38765
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