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Mirrors > Home > MPE Home > Th. List > Mathboxes > eqvreldisj4 | Structured version Visualization version GIF version |
Description: Intersection with the converse epsilon relation restricted to the quotient set of an equivalence relation is disjoint. (Contributed by Peter Mazsa, 30-May-2020.) (Revised by Peter Mazsa, 31-Dec-2021.) |
Ref | Expression |
---|---|
eqvreldisj4 | ⊢ ( EqvRel 𝑅 → Disj (𝑆 ∩ (◡ E ↾ (𝐵 / 𝑅)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqvreldisj3 38805 | . 2 ⊢ ( EqvRel 𝑅 → Disj (◡ E ↾ (𝐵 / 𝑅))) | |
2 | disjimin 38730 | . 2 ⊢ ( Disj (◡ E ↾ (𝐵 / 𝑅)) → Disj (𝑆 ∩ (◡ E ↾ (𝐵 / 𝑅)))) | |
3 | 1, 2 | syl 17 | 1 ⊢ ( EqvRel 𝑅 → Disj (𝑆 ∩ (◡ E ↾ (𝐵 / 𝑅)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∩ cin 3949 E cep 5581 ◡ccnv 5682 ↾ cres 5685 / cqs 8740 EqvRel weqvrel 38177 Disj wdisjALTV 38194 |
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-funALTV 38661 df-disjALTV 38684 df-eldisj 38686 |
This theorem is referenced by: (None) |
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