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Theorem disjim 38285
Description: The "Divide et Aequivalere" Theorem: every disjoint relation generates equivalent cosets by the relation: generalization of the former prter1 38383, cf. eldisjim 38288. (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 38220 . . . 4 ( Disj 𝑅 ↔ (∀𝑦∃*𝑢 𝑢𝑅𝑦 ∧ Rel 𝑅))
21simplbi 496 . . 3 ( Disj 𝑅 → ∀𝑦∃*𝑢 𝑢𝑅𝑦)
3 trcoss 37986 . . 3 (∀𝑦∃*𝑢 𝑢𝑅𝑦 → ∀𝑥𝑦𝑧((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))
42, 3syl 17 . 2 ( Disj 𝑅 → ∀𝑥𝑦𝑧((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))
5 eqvrelcoss3 38122 . 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 394  wal 1531  ∃*wmo 2527   class class class wbr 5152  Rel wrel 5687  ccoss 37681   EqvRel weqvrel 37698   Disj wdisjALTV 37715
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 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433
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-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-br 5153  df-opab 5215  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-coss 37915  df-refrel 38016  df-cnvrefrel 38031  df-symrel 38048  df-trrel 38078  df-eqvrel 38089  df-disjALTV 38209
This theorem is referenced by:  disjimi  38286  detlem  38287  eldisjim  38288  eldisjim2  38289  partim2  38311
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