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Mirrors > Home > MPE Home > Th. List > Mathboxes > disjim | Structured version Visualization version GIF version |
Description: The "Divide et Aequivalere" Theorem: every disjoint relation generates equivalent cosets by the relation: generalization of the former prter1 38784, cf. eldisjim 38689. (Contributed by Peter Mazsa, 3-May-2019.) (Revised by Peter Mazsa, 17-Sep-2021.) |
Ref | Expression |
---|---|
disjim | ⊢ ( Disj 𝑅 → EqvRel ≀ 𝑅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfdisjALTV4 38621 | . . . 4 ⊢ ( Disj 𝑅 ↔ (∀𝑦∃*𝑢 𝑢𝑅𝑦 ∧ Rel 𝑅)) | |
2 | 1 | simplbi 497 | . . 3 ⊢ ( Disj 𝑅 → ∀𝑦∃*𝑢 𝑢𝑅𝑦) |
3 | trcoss 38387 | . . 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 38523 | . 2 ⊢ ( EqvRel ≀ 𝑅 ↔ ∀𝑥∀𝑦∀𝑧((𝑥 ≀ 𝑅𝑦 ∧ 𝑦 ≀ 𝑅𝑧) → 𝑥 ≀ 𝑅𝑧)) | |
6 | 4, 5 | sylibr 234 | 1 ⊢ ( Disj 𝑅 → EqvRel ≀ 𝑅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∀wal 1535 ∃*wmo 2535 class class class wbr 5169 Rel wrel 5704 ≀ ccoss 38084 EqvRel weqvrel 38101 Disj wdisjALTV 38118 |
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 2105 ax-9 2113 ax-10 2136 ax-11 2153 ax-12 2173 ax-ext 2705 ax-sep 5320 ax-nul 5327 ax-pr 5450 |
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 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2890 df-ral 3064 df-rex 3073 df-rab 3439 df-v 3484 df-dif 3973 df-un 3975 df-in 3977 df-ss 3987 df-nul 4348 df-if 4549 df-sn 4649 df-pr 4651 df-op 4655 df-br 5170 df-opab 5232 df-id 5597 df-xp 5705 df-rel 5706 df-cnv 5707 df-co 5708 df-dm 5709 df-rn 5710 df-res 5711 df-coss 38316 df-refrel 38417 df-cnvrefrel 38432 df-symrel 38449 df-trrel 38479 df-eqvrel 38490 df-disjALTV 38610 |
This theorem is referenced by: disjimi 38687 detlem 38688 eldisjim 38689 eldisjim2 38690 partim2 38712 |
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