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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 38383, cf. eldisjim 38288. (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 38220 | . . . 4 ⊢ ( Disj 𝑅 ↔ (∀𝑦∃*𝑢 𝑢𝑅𝑦 ∧ Rel 𝑅)) | |
2 | 1 | simplbi 496 | . . 3 ⊢ ( Disj 𝑅 → ∀𝑦∃*𝑢 𝑢𝑅𝑦) |
3 | trcoss 37986 | . . 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 38122 | . 2 ⊢ ( EqvRel ≀ 𝑅 ↔ ∀𝑥∀𝑦∀𝑧((𝑥 ≀ 𝑅𝑦 ∧ 𝑦 ≀ 𝑅𝑧) → 𝑥 ≀ 𝑅𝑧)) | |
6 | 4, 5 | sylibr 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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