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| Description: A quotient set is a disjoint set. (Contributed by Mario Carneiro, 10-Dec-2016.) | 
| Ref | Expression | 
|---|---|
| qsdisj2 | ⊢ (𝑅 Er 𝑋 → Disj 𝑥 ∈ (𝐴 / 𝑅)𝑥) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | simpl 482 | . . . 4 ⊢ ((𝑅 Er 𝑋 ∧ (𝑥 ∈ (𝐴 / 𝑅) ∧ 𝑦 ∈ (𝐴 / 𝑅))) → 𝑅 Er 𝑋) | |
| 2 | simprl 770 | . . . 4 ⊢ ((𝑅 Er 𝑋 ∧ (𝑥 ∈ (𝐴 / 𝑅) ∧ 𝑦 ∈ (𝐴 / 𝑅))) → 𝑥 ∈ (𝐴 / 𝑅)) | |
| 3 | simprr 772 | . . . 4 ⊢ ((𝑅 Er 𝑋 ∧ (𝑥 ∈ (𝐴 / 𝑅) ∧ 𝑦 ∈ (𝐴 / 𝑅))) → 𝑦 ∈ (𝐴 / 𝑅)) | |
| 4 | 1, 2, 3 | qsdisj 8835 | . . 3 ⊢ ((𝑅 Er 𝑋 ∧ (𝑥 ∈ (𝐴 / 𝑅) ∧ 𝑦 ∈ (𝐴 / 𝑅))) → (𝑥 = 𝑦 ∨ (𝑥 ∩ 𝑦) = ∅)) | 
| 5 | 4 | ralrimivva 3201 | . 2 ⊢ (𝑅 Er 𝑋 → ∀𝑥 ∈ (𝐴 / 𝑅)∀𝑦 ∈ (𝐴 / 𝑅)(𝑥 = 𝑦 ∨ (𝑥 ∩ 𝑦) = ∅)) | 
| 6 | id 22 | . . 3 ⊢ (𝑥 = 𝑦 → 𝑥 = 𝑦) | |
| 7 | 6 | disjor 5124 | . 2 ⊢ (Disj 𝑥 ∈ (𝐴 / 𝑅)𝑥 ↔ ∀𝑥 ∈ (𝐴 / 𝑅)∀𝑦 ∈ (𝐴 / 𝑅)(𝑥 = 𝑦 ∨ (𝑥 ∩ 𝑦) = ∅)) | 
| 8 | 5, 7 | sylibr 234 | 1 ⊢ (𝑅 Er 𝑋 → Disj 𝑥 ∈ (𝐴 / 𝑅)𝑥) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 ∨ wo 847 = wceq 1539 ∈ wcel 2107 ∀wral 3060 ∩ cin 3949 ∅c0 4332 Disj wdisj 5109 Er wer 8743 / cqs 8745 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-clab 2714 df-cleq 2728 df-clel 2815 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 4626 df-pr 4628 df-op 4632 df-disj 5110 df-br 5143 df-opab 5205 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-er 8746 df-ec 8748 df-qs 8752 | 
| This theorem is referenced by: qshash 15864 | 
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