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| Mirrors > Home > MPE Home > Th. List > Mathboxes > qsdisjALTV | Structured version Visualization version GIF version | ||
| Description: Elements of a quotient set do not overlap. (Contributed by Rodolfo Medina, 12-Oct-2010.) (Revised by Mario Carneiro, 11-Jul-2014.) (Revised by Peter Mazsa, 3-Jun-2019.) | 
| Ref | Expression | 
|---|---|
| qsdisjALTV.1 | ⊢ (𝜑 → EqvRel 𝑅) | 
| qsdisjALTV.2 | ⊢ (𝜑 → 𝐵 ∈ (𝐴 / 𝑅)) | 
| qsdisjALTV.3 | ⊢ (𝜑 → 𝐶 ∈ (𝐴 / 𝑅)) | 
| Ref | Expression | 
|---|---|
| qsdisjALTV | ⊢ (𝜑 → (𝐵 = 𝐶 ∨ (𝐵 ∩ 𝐶) = ∅)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | qsdisjALTV.2 | . 2 ⊢ (𝜑 → 𝐵 ∈ (𝐴 / 𝑅)) | |
| 2 | eqid 2736 | . . 3 ⊢ (𝐴 / 𝑅) = (𝐴 / 𝑅) | |
| 3 | eqeq1 2740 | . . . 4 ⊢ ([𝑥]𝑅 = 𝐵 → ([𝑥]𝑅 = 𝐶 ↔ 𝐵 = 𝐶)) | |
| 4 | ineq1 4212 | . . . . 5 ⊢ ([𝑥]𝑅 = 𝐵 → ([𝑥]𝑅 ∩ 𝐶) = (𝐵 ∩ 𝐶)) | |
| 5 | 4 | eqeq1d 2738 | . . . 4 ⊢ ([𝑥]𝑅 = 𝐵 → (([𝑥]𝑅 ∩ 𝐶) = ∅ ↔ (𝐵 ∩ 𝐶) = ∅)) | 
| 6 | 3, 5 | orbi12d 918 | . . 3 ⊢ ([𝑥]𝑅 = 𝐵 → (([𝑥]𝑅 = 𝐶 ∨ ([𝑥]𝑅 ∩ 𝐶) = ∅) ↔ (𝐵 = 𝐶 ∨ (𝐵 ∩ 𝐶) = ∅))) | 
| 7 | qsdisjALTV.3 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ (𝐴 / 𝑅)) | |
| 8 | eqeq2 2748 | . . . . . 6 ⊢ ([𝑦]𝑅 = 𝐶 → ([𝑥]𝑅 = [𝑦]𝑅 ↔ [𝑥]𝑅 = 𝐶)) | |
| 9 | ineq2 4213 | . . . . . . 7 ⊢ ([𝑦]𝑅 = 𝐶 → ([𝑥]𝑅 ∩ [𝑦]𝑅) = ([𝑥]𝑅 ∩ 𝐶)) | |
| 10 | 9 | eqeq1d 2738 | . . . . . 6 ⊢ ([𝑦]𝑅 = 𝐶 → (([𝑥]𝑅 ∩ [𝑦]𝑅) = ∅ ↔ ([𝑥]𝑅 ∩ 𝐶) = ∅)) | 
| 11 | 8, 10 | orbi12d 918 | . . . . 5 ⊢ ([𝑦]𝑅 = 𝐶 → (([𝑥]𝑅 = [𝑦]𝑅 ∨ ([𝑥]𝑅 ∩ [𝑦]𝑅) = ∅) ↔ ([𝑥]𝑅 = 𝐶 ∨ ([𝑥]𝑅 ∩ 𝐶) = ∅))) | 
| 12 | qsdisjALTV.1 | . . . . . . 7 ⊢ (𝜑 → EqvRel 𝑅) | |
| 13 | 12 | ad2antrr 726 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) → EqvRel 𝑅) | 
| 14 | eqvreldisj 38616 | . . . . . 6 ⊢ ( EqvRel 𝑅 → ([𝑥]𝑅 = [𝑦]𝑅 ∨ ([𝑥]𝑅 ∩ [𝑦]𝑅) = ∅)) | |
| 15 | 13, 14 | syl 17 | . . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) → ([𝑥]𝑅 = [𝑦]𝑅 ∨ ([𝑥]𝑅 ∩ [𝑦]𝑅) = ∅)) | 
| 16 | 2, 11, 15 | ectocld 8825 | . . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝐶 ∈ (𝐴 / 𝑅)) → ([𝑥]𝑅 = 𝐶 ∨ ([𝑥]𝑅 ∩ 𝐶) = ∅)) | 
| 17 | 7, 16 | mpidan 689 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ([𝑥]𝑅 = 𝐶 ∨ ([𝑥]𝑅 ∩ 𝐶) = ∅)) | 
| 18 | 2, 6, 17 | ectocld 8825 | . 2 ⊢ ((𝜑 ∧ 𝐵 ∈ (𝐴 / 𝑅)) → (𝐵 = 𝐶 ∨ (𝐵 ∩ 𝐶) = ∅)) | 
| 19 | 1, 18 | mpdan 687 | 1 ⊢ (𝜑 → (𝐵 = 𝐶 ∨ (𝐵 ∩ 𝐶) = ∅)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 ∨ wo 847 = wceq 1539 ∈ wcel 2107 ∩ cin 3949 ∅c0 4332 [cec 8744 / cqs 8745 EqvRel weqvrel 38200 | 
| 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-clab 2714 df-cleq 2728 df-clel 2815 df-ne 2940 df-ral 3061 df-rex 3070 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-br 5143 df-opab 5205 df-id 5577 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-ec 8748 df-qs 8752 df-refrel 38514 df-symrel 38546 df-trrel 38576 df-eqvrel 38587 | 
| This theorem is referenced by: eqvreldisj1 38826 | 
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