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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 38353). (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 38351 | . 2 ⊢ ( EqvRel 𝑅 → ∀𝑥 ∈ (𝐴 / 𝑅)∀𝑦 ∈ (𝐴 / 𝑅)(𝑥 = 𝑦 ∨ (𝑥 ∩ 𝑦) = ∅)) | |
2 | dfeldisj5 38248 | . 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 845 = wceq 1533 ∀wral 3051 ∩ cin 3939 ∅c0 4318 / cqs 8720 EqvRel weqvrel 37721 ElDisj weldisj 37740 |
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 2696 ax-sep 5294 ax-nul 5301 ax-pr 5423 |
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 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-ral 3052 df-rex 3061 df-rmo 3364 df-rab 3420 df-v 3465 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-nul 4319 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-br 5144 df-opab 5206 df-id 5570 df-eprel 5576 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-ec 8723 df-qs 8727 df-coss 37938 df-refrel 38039 df-cnvrefrel 38054 df-symrel 38071 df-trrel 38101 df-eqvrel 38112 df-disjALTV 38232 df-eldisj 38234 |
This theorem is referenced by: eqvreldisj3 38353 eqvrelqseqdisj2 38356 |
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