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Mirrors > Home > MPE Home > Th. List > qsdisj | Structured version Visualization version GIF version |
Description: Members of a quotient set do not overlap. (Contributed by Rodolfo Medina, 12-Oct-2010.) (Revised by Mario Carneiro, 11-Jul-2014.) |
Ref | Expression |
---|---|
qsdisj.1 | ⊢ (𝜑 → 𝑅 Er 𝑋) |
qsdisj.2 | ⊢ (𝜑 → 𝐵 ∈ (𝐴 / 𝑅)) |
qsdisj.3 | ⊢ (𝜑 → 𝐶 ∈ (𝐴 / 𝑅)) |
Ref | Expression |
---|---|
qsdisj | ⊢ (𝜑 → (𝐵 = 𝐶 ∨ (𝐵 ∩ 𝐶) = ∅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | qsdisj.2 | . 2 ⊢ (𝜑 → 𝐵 ∈ (𝐴 / 𝑅)) | |
2 | eqid 2798 | . . 3 ⊢ (𝐴 / 𝑅) = (𝐴 / 𝑅) | |
3 | eqeq1 2802 | . . . 4 ⊢ ([𝑥]𝑅 = 𝐵 → ([𝑥]𝑅 = 𝐶 ↔ 𝐵 = 𝐶)) | |
4 | ineq1 4131 | . . . . 5 ⊢ ([𝑥]𝑅 = 𝐵 → ([𝑥]𝑅 ∩ 𝐶) = (𝐵 ∩ 𝐶)) | |
5 | 4 | eqeq1d 2800 | . . . 4 ⊢ ([𝑥]𝑅 = 𝐵 → (([𝑥]𝑅 ∩ 𝐶) = ∅ ↔ (𝐵 ∩ 𝐶) = ∅)) |
6 | 3, 5 | orbi12d 916 | . . 3 ⊢ ([𝑥]𝑅 = 𝐵 → (([𝑥]𝑅 = 𝐶 ∨ ([𝑥]𝑅 ∩ 𝐶) = ∅) ↔ (𝐵 = 𝐶 ∨ (𝐵 ∩ 𝐶) = ∅))) |
7 | qsdisj.3 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ (𝐴 / 𝑅)) | |
8 | eqeq2 2810 | . . . . . 6 ⊢ ([𝑦]𝑅 = 𝐶 → ([𝑥]𝑅 = [𝑦]𝑅 ↔ [𝑥]𝑅 = 𝐶)) | |
9 | ineq2 4133 | . . . . . . 7 ⊢ ([𝑦]𝑅 = 𝐶 → ([𝑥]𝑅 ∩ [𝑦]𝑅) = ([𝑥]𝑅 ∩ 𝐶)) | |
10 | 9 | eqeq1d 2800 | . . . . . 6 ⊢ ([𝑦]𝑅 = 𝐶 → (([𝑥]𝑅 ∩ [𝑦]𝑅) = ∅ ↔ ([𝑥]𝑅 ∩ 𝐶) = ∅)) |
11 | 8, 10 | orbi12d 916 | . . . . 5 ⊢ ([𝑦]𝑅 = 𝐶 → (([𝑥]𝑅 = [𝑦]𝑅 ∨ ([𝑥]𝑅 ∩ [𝑦]𝑅) = ∅) ↔ ([𝑥]𝑅 = 𝐶 ∨ ([𝑥]𝑅 ∩ 𝐶) = ∅))) |
12 | qsdisj.1 | . . . . . . 7 ⊢ (𝜑 → 𝑅 Er 𝑋) | |
13 | 12 | ad2antrr 725 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) → 𝑅 Er 𝑋) |
14 | erdisj 8324 | . . . . . 6 ⊢ (𝑅 Er 𝑋 → ([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 8347 | . . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝐶 ∈ (𝐴 / 𝑅)) → ([𝑥]𝑅 = 𝐶 ∨ ([𝑥]𝑅 ∩ 𝐶) = ∅)) |
17 | 7, 16 | mpidan 688 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ([𝑥]𝑅 = 𝐶 ∨ ([𝑥]𝑅 ∩ 𝐶) = ∅)) |
18 | 2, 6, 17 | ectocld 8347 | . 2 ⊢ ((𝜑 ∧ 𝐵 ∈ (𝐴 / 𝑅)) → (𝐵 = 𝐶 ∨ (𝐵 ∩ 𝐶) = ∅)) |
19 | 1, 18 | mpdan 686 | 1 ⊢ (𝜑 → (𝐵 = 𝐶 ∨ (𝐵 ∩ 𝐶) = ∅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∨ wo 844 = wceq 1538 ∈ wcel 2111 ∩ cin 3880 ∅c0 4243 Er wer 8269 [cec 8270 / cqs 8271 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-br 5031 df-opab 5093 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-er 8272 df-ec 8274 df-qs 8278 |
This theorem is referenced by: qsdisj2 8358 uniinqs 8360 cldsubg 22716 erprt 36169 |
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