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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 8831). (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 38804 | . 2 ⊢ ( EqvRel 𝑅 → ElDisj (𝐴 / 𝑅)) | |
| 2 | df-eldisj 38686 | . 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 8740 EqvRel weqvrel 38177 Disj wdisjALTV 38194 ElDisj weldisj 38196 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 ax-sep 5294 ax-nul 5304 ax-pr 5430 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-rmo 3379 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-br 5142 df-opab 5204 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 8743 df-qs 8747 df-coss 38390 df-refrel 38491 df-cnvrefrel 38506 df-symrel 38523 df-trrel 38553 df-eqvrel 38564 df-disjALTV 38684 df-eldisj 38686 |
| This theorem is referenced by: eqvreldisj4 38806 eqvreldisj5 38807 eqvrelqseqdisj3 38810 |
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