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Mirrors > Home > MPE Home > Th. List > Mathboxes > disjlem14 | Structured version Visualization version GIF version |
Description: Lemma for disjdmqseq 38188, partim2 38190 and petlem 38195 via disjlem17 38182, (general version of the former prtlem14 38257). (Contributed by Peter Mazsa, 10-Sep-2021.) |
Ref | Expression |
---|---|
disjlem14 | ⊢ ( Disj 𝑅 → ((𝑥 ∈ dom 𝑅 ∧ 𝑦 ∈ dom 𝑅) → ((𝐴 ∈ [𝑥]𝑅 ∧ 𝐴 ∈ [𝑦]𝑅) → [𝑥]𝑅 = [𝑦]𝑅))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfdisjALTV5 38100 | . . . 4 ⊢ ( Disj 𝑅 ↔ (∀𝑥 ∈ dom 𝑅∀𝑦 ∈ dom 𝑅(𝑥 = 𝑦 ∨ ([𝑥]𝑅 ∩ [𝑦]𝑅) = ∅) ∧ Rel 𝑅)) | |
2 | 1 | simplbi 497 | . . 3 ⊢ ( Disj 𝑅 → ∀𝑥 ∈ dom 𝑅∀𝑦 ∈ dom 𝑅(𝑥 = 𝑦 ∨ ([𝑥]𝑅 ∩ [𝑦]𝑅) = ∅)) |
3 | rsp2 3268 | . . 3 ⊢ (∀𝑥 ∈ dom 𝑅∀𝑦 ∈ dom 𝑅(𝑥 = 𝑦 ∨ ([𝑥]𝑅 ∩ [𝑦]𝑅) = ∅) → ((𝑥 ∈ dom 𝑅 ∧ 𝑦 ∈ dom 𝑅) → (𝑥 = 𝑦 ∨ ([𝑥]𝑅 ∩ [𝑦]𝑅) = ∅))) | |
4 | 2, 3 | syl 17 | . 2 ⊢ ( Disj 𝑅 → ((𝑥 ∈ dom 𝑅 ∧ 𝑦 ∈ dom 𝑅) → (𝑥 = 𝑦 ∨ ([𝑥]𝑅 ∩ [𝑦]𝑅) = ∅))) |
5 | eceq1 8743 | . . . 4 ⊢ (𝑥 = 𝑦 → [𝑥]𝑅 = [𝑦]𝑅) | |
6 | 5 | a1d 25 | . . 3 ⊢ (𝑥 = 𝑦 → ((𝐴 ∈ [𝑥]𝑅 ∧ 𝐴 ∈ [𝑦]𝑅) → [𝑥]𝑅 = [𝑦]𝑅)) |
7 | elin 3959 | . . . 4 ⊢ (𝐴 ∈ ([𝑥]𝑅 ∩ [𝑦]𝑅) ↔ (𝐴 ∈ [𝑥]𝑅 ∧ 𝐴 ∈ [𝑦]𝑅)) | |
8 | nel02 4327 | . . . . 5 ⊢ (([𝑥]𝑅 ∩ [𝑦]𝑅) = ∅ → ¬ 𝐴 ∈ ([𝑥]𝑅 ∩ [𝑦]𝑅)) | |
9 | 8 | pm2.21d 121 | . . . 4 ⊢ (([𝑥]𝑅 ∩ [𝑦]𝑅) = ∅ → (𝐴 ∈ ([𝑥]𝑅 ∩ [𝑦]𝑅) → [𝑥]𝑅 = [𝑦]𝑅)) |
10 | 7, 9 | biimtrrid 242 | . . 3 ⊢ (([𝑥]𝑅 ∩ [𝑦]𝑅) = ∅ → ((𝐴 ∈ [𝑥]𝑅 ∧ 𝐴 ∈ [𝑦]𝑅) → [𝑥]𝑅 = [𝑦]𝑅)) |
11 | 6, 10 | jaoi 854 | . 2 ⊢ ((𝑥 = 𝑦 ∨ ([𝑥]𝑅 ∩ [𝑦]𝑅) = ∅) → ((𝐴 ∈ [𝑥]𝑅 ∧ 𝐴 ∈ [𝑦]𝑅) → [𝑥]𝑅 = [𝑦]𝑅)) |
12 | 4, 11 | syl6 35 | 1 ⊢ ( Disj 𝑅 → ((𝑥 ∈ dom 𝑅 ∧ 𝑦 ∈ dom 𝑅) → ((𝐴 ∈ [𝑥]𝑅 ∧ 𝐴 ∈ [𝑦]𝑅) → [𝑥]𝑅 = [𝑦]𝑅))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∨ wo 844 = wceq 1533 ∈ wcel 2098 ∀wral 3055 ∩ cin 3942 ∅c0 4317 dom cdm 5669 Rel wrel 5674 [cec 8703 Disj wdisjALTV 37590 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pr 5420 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ral 3056 df-rex 3065 df-rmo 3370 df-rab 3427 df-v 3470 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-br 5142 df-opab 5204 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-ec 8707 df-coss 37794 df-cnvrefrel 37910 df-disjALTV 38088 |
This theorem is referenced by: disjlem17 38182 |
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