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Mirrors > Home > MPE Home > Th. List > Mathboxes > eqvreldisj2 | Structured version Visualization version GIF version |
Description: The elements of the quotient set of an equivalence relation are disjoint (cf. eqvreldisj3 38209). (Contributed by Mario Carneiro, 10-Dec-2016.) (Revised by Peter Mazsa, 19-Sep-2021.) |
Ref | Expression |
---|---|
eqvreldisj2 | ⊢ ( EqvRel 𝑅 → ElDisj (𝐴 / 𝑅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqvreldisj1 38207 | . 2 ⊢ ( EqvRel 𝑅 → ∀𝑥 ∈ (𝐴 / 𝑅)∀𝑦 ∈ (𝐴 / 𝑅)(𝑥 = 𝑦 ∨ (𝑥 ∩ 𝑦) = ∅)) | |
2 | dfeldisj5 38104 | . 2 ⊢ ( ElDisj (𝐴 / 𝑅) ↔ ∀𝑥 ∈ (𝐴 / 𝑅)∀𝑦 ∈ (𝐴 / 𝑅)(𝑥 = 𝑦 ∨ (𝑥 ∩ 𝑦) = ∅)) | |
3 | 1, 2 | sylibr 233 | 1 ⊢ ( EqvRel 𝑅 → ElDisj (𝐴 / 𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∨ wo 844 = wceq 1533 ∀wral 3055 ∩ cin 3942 ∅c0 4317 / cqs 8704 EqvRel weqvrel 37573 ElDisj weldisj 37592 |
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 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pr 5420 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-rmo 3370 df-rab 3427 df-v 3470 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-br 5142 df-opab 5204 df-id 5567 df-eprel 5573 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-ec 8707 df-qs 8711 df-coss 37794 df-refrel 37895 df-cnvrefrel 37910 df-symrel 37927 df-trrel 37957 df-eqvrel 37968 df-disjALTV 38088 df-eldisj 38090 |
This theorem is referenced by: eqvreldisj3 38209 eqvrelqseqdisj2 38212 |
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