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Mirrors > Home > MPE Home > Th. List > Mathboxes > eqvreldisj3 | Structured version Visualization version GIF version |
Description: The elements of the quotient set of an equivalence relation are disjoint (cf. qsdisj2 8828). (Contributed by Mario Carneiro, 10-Dec-2016.) (Revised by Peter Mazsa, 20-Jun-2019.) (Revised by Peter Mazsa, 19-Sep-2021.) |
Ref | Expression |
---|---|
eqvreldisj3 | ⊢ ( EqvRel 𝑅 → Disj (◡ E ↾ (𝐴 / 𝑅))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqvreldisj2 38768 | . 2 ⊢ ( EqvRel 𝑅 → ElDisj (𝐴 / 𝑅)) | |
2 | df-eldisj 38650 | . 2 ⊢ ( ElDisj (𝐴 / 𝑅) ↔ Disj (◡ E ↾ (𝐴 / 𝑅))) | |
3 | 1, 2 | sylib 218 | 1 ⊢ ( EqvRel 𝑅 → Disj (◡ E ↾ (𝐴 / 𝑅))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 E cep 5581 ◡ccnv 5682 ↾ cres 5685 / cqs 8737 EqvRel weqvrel 38139 Disj wdisjALTV 38156 ElDisj weldisj 38158 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1963 ax-7 2003 ax-8 2106 ax-9 2114 ax-10 2137 ax-11 2153 ax-12 2173 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pr 5430 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3an 1087 df-tru 1538 df-fal 1548 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2536 df-eu 2565 df-clab 2711 df-cleq 2725 df-clel 2812 df-nfc 2888 df-ne 2937 df-ral 3058 df-rex 3067 df-rmo 3376 df-rab 3433 df-v 3479 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-br 5150 df-opab 5212 df-id 5576 df-eprel 5582 df-xp 5689 df-rel 5690 df-cnv 5691 df-co 5692 df-dm 5693 df-rn 5694 df-res 5695 df-ima 5696 df-ec 8740 df-qs 8744 df-coss 38354 df-refrel 38455 df-cnvrefrel 38470 df-symrel 38487 df-trrel 38517 df-eqvrel 38528 df-disjALTV 38648 df-eldisj 38650 |
This theorem is referenced by: eqvreldisj4 38770 eqvreldisj5 38771 eqvrelqseqdisj3 38774 |
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