Theorem disjim 38899

Description: The "Divide et Aequivalere" Theorem: every disjoint relation generates equivalent cosets by the relation: generalization of the former prter1 38998, cf. eldisjim 38902. (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 38834	. . . 4 ⊢ ( Disj 𝑅 ↔ (∀𝑦∃*𝑢 𝑢𝑅𝑦 ∧ Rel 𝑅))
2	1	simplbi 497	. . 3 ⊢ ( Disj 𝑅 → ∀𝑦∃*𝑢 𝑢𝑅𝑦)
3		trcoss 38604	. . 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 38734	. 2 ⊢ ( EqvRel ≀ 𝑅 ↔ ∀𝑥∀𝑦∀𝑧((𝑥 ≀ 𝑅𝑦 ∧ 𝑦 ≀ 𝑅𝑧) → 𝑥 ≀ 𝑅𝑧))
6	4, 5	sylibr 234	1 ⊢ ( Disj 𝑅 → EqvRel ≀ 𝑅)

Syntax hints:   → wi 4   ∧ wa 395  ∀wal 1539  ∃*wmo 2535   class class class wbr 5093  Rel wrel 5624   ≀ ccoss 38242   EqvRel weqvrel 38259   Disj wdisjALTV 38276

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-sep 5236  ax-nul 5246  ax-pr 5372

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4475  df-sn 4576  df-pr 4578  df-op 4582  df-br 5094  df-opab 5156  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-coss 38533  df-refrel 38624  df-cnvrefrel 38639  df-symrel 38656  df-trrel 38690  df-eqvrel 38701  df-disjALTV 38823

This theorem is referenced by:  disjimi  38900  detlem  38901  eldisjim  38902  eldisjim2  38903  partim2  38925