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Mirrors > Home > ILE Home > Th. List > dedekindeulemlub | GIF version |
Description: Lemma for dedekindeu 13148. The set L has a least upper bound. (Contributed by Jim Kingdon, 31-Jan-2024.) |
Ref | Expression |
---|---|
dedekindeu.lss | ⊢ (𝜑 → 𝐿 ⊆ ℝ) |
dedekindeu.uss | ⊢ (𝜑 → 𝑈 ⊆ ℝ) |
dedekindeu.lm | ⊢ (𝜑 → ∃𝑞 ∈ ℝ 𝑞 ∈ 𝐿) |
dedekindeu.um | ⊢ (𝜑 → ∃𝑟 ∈ ℝ 𝑟 ∈ 𝑈) |
dedekindeu.lr | ⊢ (𝜑 → ∀𝑞 ∈ ℝ (𝑞 ∈ 𝐿 ↔ ∃𝑟 ∈ 𝐿 𝑞 < 𝑟)) |
dedekindeu.ur | ⊢ (𝜑 → ∀𝑟 ∈ ℝ (𝑟 ∈ 𝑈 ↔ ∃𝑞 ∈ 𝑈 𝑞 < 𝑟)) |
dedekindeu.disj | ⊢ (𝜑 → (𝐿 ∩ 𝑈) = ∅) |
dedekindeu.loc | ⊢ (𝜑 → ∀𝑞 ∈ ℝ ∀𝑟 ∈ ℝ (𝑞 < 𝑟 → (𝑞 ∈ 𝐿 ∨ 𝑟 ∈ 𝑈))) |
Ref | Expression |
---|---|
dedekindeulemlub | ⊢ (𝜑 → ∃𝑥 ∈ ℝ (∀𝑦 ∈ 𝐿 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐿 𝑦 < 𝑧))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dedekindeu.lss | . 2 ⊢ (𝜑 → 𝐿 ⊆ ℝ) | |
2 | dedekindeu.lm | . . 3 ⊢ (𝜑 → ∃𝑞 ∈ ℝ 𝑞 ∈ 𝐿) | |
3 | eleq1w 2225 | . . . . 5 ⊢ (𝑞 = 𝑥 → (𝑞 ∈ 𝐿 ↔ 𝑥 ∈ 𝐿)) | |
4 | 3 | cbvrexv 2690 | . . . 4 ⊢ (∃𝑞 ∈ ℝ 𝑞 ∈ 𝐿 ↔ ∃𝑥 ∈ ℝ 𝑥 ∈ 𝐿) |
5 | rexex 2510 | . . . 4 ⊢ (∃𝑥 ∈ ℝ 𝑥 ∈ 𝐿 → ∃𝑥 𝑥 ∈ 𝐿) | |
6 | 4, 5 | sylbi 120 | . . 3 ⊢ (∃𝑞 ∈ ℝ 𝑞 ∈ 𝐿 → ∃𝑥 𝑥 ∈ 𝐿) |
7 | 2, 6 | syl 14 | . 2 ⊢ (𝜑 → ∃𝑥 𝑥 ∈ 𝐿) |
8 | dedekindeu.uss | . . 3 ⊢ (𝜑 → 𝑈 ⊆ ℝ) | |
9 | dedekindeu.um | . . 3 ⊢ (𝜑 → ∃𝑟 ∈ ℝ 𝑟 ∈ 𝑈) | |
10 | dedekindeu.lr | . . 3 ⊢ (𝜑 → ∀𝑞 ∈ ℝ (𝑞 ∈ 𝐿 ↔ ∃𝑟 ∈ 𝐿 𝑞 < 𝑟)) | |
11 | dedekindeu.ur | . . 3 ⊢ (𝜑 → ∀𝑟 ∈ ℝ (𝑟 ∈ 𝑈 ↔ ∃𝑞 ∈ 𝑈 𝑞 < 𝑟)) | |
12 | dedekindeu.disj | . . 3 ⊢ (𝜑 → (𝐿 ∩ 𝑈) = ∅) | |
13 | dedekindeu.loc | . . 3 ⊢ (𝜑 → ∀𝑞 ∈ ℝ ∀𝑟 ∈ ℝ (𝑞 < 𝑟 → (𝑞 ∈ 𝐿 ∨ 𝑟 ∈ 𝑈))) | |
14 | 1, 8, 2, 9, 10, 11, 12, 13 | dedekindeulemub 13143 | . 2 ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐿 𝑦 < 𝑥) |
15 | 1, 8, 2, 9, 10, 11, 12, 13 | dedekindeulemloc 13144 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 → (∃𝑧 ∈ 𝐿 𝑥 < 𝑧 ∨ ∀𝑧 ∈ 𝐿 𝑧 < 𝑦))) |
16 | axsuploc 7962 | . 2 ⊢ (((𝐿 ⊆ ℝ ∧ ∃𝑥 𝑥 ∈ 𝐿) ∧ (∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐿 𝑦 < 𝑥 ∧ ∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 → (∃𝑧 ∈ 𝐿 𝑥 < 𝑧 ∨ ∀𝑧 ∈ 𝐿 𝑧 < 𝑦)))) → ∃𝑥 ∈ ℝ (∀𝑦 ∈ 𝐿 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐿 𝑦 < 𝑧))) | |
17 | 1, 7, 14, 15, 16 | syl22anc 1228 | 1 ⊢ (𝜑 → ∃𝑥 ∈ ℝ (∀𝑦 ∈ 𝐿 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐿 𝑦 < 𝑧))) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 103 ↔ wb 104 ∨ wo 698 = wceq 1342 ∃wex 1479 ∈ wcel 2135 ∀wral 2442 ∃wrex 2443 ∩ cin 3110 ⊆ wss 3111 ∅c0 3404 class class class wbr 3976 ℝcr 7743 < clt 7924 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 ax-cnex 7835 ax-resscn 7836 ax-pre-ltwlin 7857 ax-pre-suploc 7865 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-nel 2430 df-ral 2447 df-rex 2448 df-rab 2451 df-v 2723 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-nul 3405 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-br 3977 df-opab 4038 df-xp 4604 df-cnv 4606 df-pnf 7926 df-mnf 7927 df-xr 7928 df-ltxr 7929 df-le 7930 |
This theorem is referenced by: dedekindeulemlu 13146 |
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