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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 2820 | . . 3 ⊢ (𝐴 / 𝑅) = (𝐴 / 𝑅) | |
3 | eqeq1 2824 | . . . 4 ⊢ ([𝑥]𝑅 = 𝐵 → ([𝑥]𝑅 = 𝐶 ↔ 𝐵 = 𝐶)) | |
4 | ineq1 4174 | . . . . 5 ⊢ ([𝑥]𝑅 = 𝐵 → ([𝑥]𝑅 ∩ 𝐶) = (𝐵 ∩ 𝐶)) | |
5 | 4 | eqeq1d 2822 | . . . 4 ⊢ ([𝑥]𝑅 = 𝐵 → (([𝑥]𝑅 ∩ 𝐶) = ∅ ↔ (𝐵 ∩ 𝐶) = ∅)) |
6 | 3, 5 | orbi12d 915 | . . 3 ⊢ ([𝑥]𝑅 = 𝐵 → (([𝑥]𝑅 = 𝐶 ∨ ([𝑥]𝑅 ∩ 𝐶) = ∅) ↔ (𝐵 = 𝐶 ∨ (𝐵 ∩ 𝐶) = ∅))) |
7 | qsdisjALTV.3 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ (𝐴 / 𝑅)) | |
8 | eqeq2 2832 | . . . . . 6 ⊢ ([𝑦]𝑅 = 𝐶 → ([𝑥]𝑅 = [𝑦]𝑅 ↔ [𝑥]𝑅 = 𝐶)) | |
9 | ineq2 4176 | . . . . . . 7 ⊢ ([𝑦]𝑅 = 𝐶 → ([𝑥]𝑅 ∩ [𝑦]𝑅) = ([𝑥]𝑅 ∩ 𝐶)) | |
10 | 9 | eqeq1d 2822 | . . . . . 6 ⊢ ([𝑦]𝑅 = 𝐶 → (([𝑥]𝑅 ∩ [𝑦]𝑅) = ∅ ↔ ([𝑥]𝑅 ∩ 𝐶) = ∅)) |
11 | 8, 10 | orbi12d 915 | . . . . 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 724 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) → EqvRel 𝑅) |
14 | eqvreldisj 35883 | . . . . . 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 8357 | . . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝐶 ∈ (𝐴 / 𝑅)) → ([𝑥]𝑅 = 𝐶 ∨ ([𝑥]𝑅 ∩ 𝐶) = ∅)) |
17 | 7, 16 | mpidan 687 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ([𝑥]𝑅 = 𝐶 ∨ ([𝑥]𝑅 ∩ 𝐶) = ∅)) |
18 | 2, 6, 17 | ectocld 8357 | . 2 ⊢ ((𝜑 ∧ 𝐵 ∈ (𝐴 / 𝑅)) → (𝐵 = 𝐶 ∨ (𝐵 ∩ 𝐶) = ∅)) |
19 | 1, 18 | mpdan 685 | 1 ⊢ (𝜑 → (𝐵 = 𝐶 ∨ (𝐵 ∩ 𝐶) = ∅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∨ wo 843 = wceq 1536 ∈ wcel 2113 ∩ cin 3928 ∅c0 4284 [cec 8280 / cqs 8281 EqvRel weqvrel 35504 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-sep 5196 ax-nul 5203 ax-pr 5323 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-ral 3142 df-rex 3143 df-rab 3146 df-v 3493 df-sbc 3769 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-nul 4285 df-if 4461 df-sn 4561 df-pr 4563 df-op 4567 df-br 5060 df-opab 5122 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-ec 8284 df-qs 8288 df-refrel 35786 df-symrel 35814 df-trrel 35844 df-eqvrel 35854 |
This theorem is referenced by: (None) |
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