Theorem disjim 39258

Description: The "Divide et Aequivalere" Theorem: every disjoint relation generates equivalent cosets by the relation: generalization of the former prter1 39378, cf. eldisjim 39261. (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.

Step	Hyp	Ref	Expression
1		dfdisjALTV4 39175	. . . 4 ⊢ ( Disj 𝑅 ↔ (∀𝑦∃*𝑢 𝑢𝑅𝑦 ∧ Rel 𝑅))
2	1	simplbi 497	. . 3 ⊢ ( Disj 𝑅 → ∀𝑦∃*𝑢 𝑢𝑅𝑦)
3		trcoss 38946	. . 3 ⊢ (∀𝑦∃*𝑢 𝑢𝑅𝑦 → ∀𝑥∀𝑦∀𝑧((𝑥 ≀ 𝑅𝑦 ∧ 𝑦 ≀ 𝑅𝑧) → 𝑥 ≀ 𝑅𝑧))
4	2, 3	syl 17	. 2 ⊢ ( Disj 𝑅 → ∀𝑥∀𝑦∀𝑧((𝑥 ≀ 𝑅𝑦 ∧ 𝑦 ≀ 𝑅𝑧) → 𝑥 ≀ 𝑅𝑧))
5		eqvrelcoss3 39076	. 2 ⊢ ( EqvRel ≀ 𝑅 ↔ ∀𝑥∀𝑦∀𝑧((𝑥 ≀ 𝑅𝑦 ∧ 𝑦 ≀ 𝑅𝑧) → 𝑥 ≀ 𝑅𝑧))
6	4, 5	sylibr 235	1 ⊢ ( Disj 𝑅 → EqvRel ≀ 𝑅)

Syntax hints:   → wi 4   ∧ wa 396  ∀wal 1545  ∃*wmo 2541   class class class wbr 5079  Rel wrel 5630   ≀ ccoss 38557   EqvRel weqvrel 38574   Disj wdisjALTV 38593

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-sep 5225  ax-pr 5369

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ral 3055  df-rex 3065  df-rab 3393  df-v 3434  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-sn 4563  df-pr 4565  df-op 4569  df-br 5080  df-opab 5142  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-coss 38875  df-refrel 38966  df-cnvrefrel 38981  df-symrel 38998  df-trrel 39032  df-eqvrel 39043  df-disjALTV 39164

This theorem is referenced by:  disjimi  39259  detlem  39260  eldisjim  39261  eldisjim2  39262  partim2  39284  disjimeldisjdmqs  39307