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Mirrors > Home > ILE Home > Th. List > dedekindeulemlub | GIF version |
Description: Lemma for dedekindeu 12773. 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 2200 | . . . . 5 ⊢ (𝑞 = 𝑥 → (𝑞 ∈ 𝐿 ↔ 𝑥 ∈ 𝐿)) | |
4 | 3 | cbvrexv 2655 | . . . 4 ⊢ (∃𝑞 ∈ ℝ 𝑞 ∈ 𝐿 ↔ ∃𝑥 ∈ ℝ 𝑥 ∈ 𝐿) |
5 | rexex 2479 | . . . 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 12768 | . 2 ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐿 𝑦 < 𝑥) |
15 | 1, 8, 2, 9, 10, 11, 12, 13 | dedekindeulemloc 12769 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 → (∃𝑧 ∈ 𝐿 𝑥 < 𝑧 ∨ ∀𝑧 ∈ 𝐿 𝑧 < 𝑦))) |
16 | axsuploc 7840 | . 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 1217 | 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 697 = wceq 1331 ∃wex 1468 ∈ wcel 1480 ∀wral 2416 ∃wrex 2417 ∩ cin 3070 ⊆ wss 3071 ∅c0 3363 class class class wbr 3929 ℝcr 7622 < clt 7803 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-cnex 7714 ax-resscn 7715 ax-pre-ltwlin 7736 ax-pre-suploc 7744 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-rab 2425 df-v 2688 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-xp 4545 df-cnv 4547 df-pnf 7805 df-mnf 7806 df-xr 7807 df-ltxr 7808 df-le 7809 |
This theorem is referenced by: dedekindeulemlu 12771 |
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