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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 2735 | . . 3 ⊢ (𝐴 / 𝑅) = (𝐴 / 𝑅) | |
3 | eqeq1 2739 | . . . 4 ⊢ ([𝑥]𝑅 = 𝐵 → ([𝑥]𝑅 = 𝐶 ↔ 𝐵 = 𝐶)) | |
4 | ineq1 4221 | . . . . 5 ⊢ ([𝑥]𝑅 = 𝐵 → ([𝑥]𝑅 ∩ 𝐶) = (𝐵 ∩ 𝐶)) | |
5 | 4 | eqeq1d 2737 | . . . 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 2747 | . . . . . 6 ⊢ ([𝑦]𝑅 = 𝐶 → ([𝑥]𝑅 = [𝑦]𝑅 ↔ [𝑥]𝑅 = 𝐶)) | |
9 | ineq2 4222 | . . . . . . 7 ⊢ ([𝑦]𝑅 = 𝐶 → ([𝑥]𝑅 ∩ [𝑦]𝑅) = ([𝑥]𝑅 ∩ 𝐶)) | |
10 | 9 | eqeq1d 2737 | . . . . . 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 38596 | . . . . . 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 8823 | . . . 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 8823 | . 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 1537 ∈ wcel 2106 ∩ cin 3962 ∅c0 4339 [cec 8742 / cqs 8743 EqvRel weqvrel 38179 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-br 5149 df-opab 5211 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-ec 8746 df-qs 8750 df-refrel 38494 df-symrel 38526 df-trrel 38556 df-eqvrel 38567 |
This theorem is referenced by: eqvreldisj1 38806 |
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