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Mirrors > Home > ILE Home > Th. List > dedekindeulemub | GIF version |
Description: Lemma for dedekindeu 14372. The lower cut has an 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 |
---|---|
dedekindeulemub | ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐿 𝑦 < 𝑥) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dedekindeu.um | . . 3 ⊢ (𝜑 → ∃𝑟 ∈ ℝ 𝑟 ∈ 𝑈) | |
2 | eleq1w 2248 | . . . 4 ⊢ (𝑟 = 𝑎 → (𝑟 ∈ 𝑈 ↔ 𝑎 ∈ 𝑈)) | |
3 | 2 | cbvrexv 2716 | . . 3 ⊢ (∃𝑟 ∈ ℝ 𝑟 ∈ 𝑈 ↔ ∃𝑎 ∈ ℝ 𝑎 ∈ 𝑈) |
4 | 1, 3 | sylib 122 | . 2 ⊢ (𝜑 → ∃𝑎 ∈ ℝ 𝑎 ∈ 𝑈) |
5 | simprl 529 | . . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ ℝ ∧ 𝑎 ∈ 𝑈)) → 𝑎 ∈ ℝ) | |
6 | dedekindeu.lss | . . . . 5 ⊢ (𝜑 → 𝐿 ⊆ ℝ) | |
7 | 6 | adantr 276 | . . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ ℝ ∧ 𝑎 ∈ 𝑈)) → 𝐿 ⊆ ℝ) |
8 | dedekindeu.uss | . . . . 5 ⊢ (𝜑 → 𝑈 ⊆ ℝ) | |
9 | 8 | adantr 276 | . . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ ℝ ∧ 𝑎 ∈ 𝑈)) → 𝑈 ⊆ ℝ) |
10 | dedekindeu.lm | . . . . 5 ⊢ (𝜑 → ∃𝑞 ∈ ℝ 𝑞 ∈ 𝐿) | |
11 | 10 | adantr 276 | . . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ ℝ ∧ 𝑎 ∈ 𝑈)) → ∃𝑞 ∈ ℝ 𝑞 ∈ 𝐿) |
12 | 1 | adantr 276 | . . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ ℝ ∧ 𝑎 ∈ 𝑈)) → ∃𝑟 ∈ ℝ 𝑟 ∈ 𝑈) |
13 | dedekindeu.lr | . . . . 5 ⊢ (𝜑 → ∀𝑞 ∈ ℝ (𝑞 ∈ 𝐿 ↔ ∃𝑟 ∈ 𝐿 𝑞 < 𝑟)) | |
14 | 13 | adantr 276 | . . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ ℝ ∧ 𝑎 ∈ 𝑈)) → ∀𝑞 ∈ ℝ (𝑞 ∈ 𝐿 ↔ ∃𝑟 ∈ 𝐿 𝑞 < 𝑟)) |
15 | dedekindeu.ur | . . . . 5 ⊢ (𝜑 → ∀𝑟 ∈ ℝ (𝑟 ∈ 𝑈 ↔ ∃𝑞 ∈ 𝑈 𝑞 < 𝑟)) | |
16 | 15 | adantr 276 | . . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ ℝ ∧ 𝑎 ∈ 𝑈)) → ∀𝑟 ∈ ℝ (𝑟 ∈ 𝑈 ↔ ∃𝑞 ∈ 𝑈 𝑞 < 𝑟)) |
17 | dedekindeu.disj | . . . . 5 ⊢ (𝜑 → (𝐿 ∩ 𝑈) = ∅) | |
18 | 17 | adantr 276 | . . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ ℝ ∧ 𝑎 ∈ 𝑈)) → (𝐿 ∩ 𝑈) = ∅) |
19 | dedekindeu.loc | . . . . 5 ⊢ (𝜑 → ∀𝑞 ∈ ℝ ∀𝑟 ∈ ℝ (𝑞 < 𝑟 → (𝑞 ∈ 𝐿 ∨ 𝑟 ∈ 𝑈))) | |
20 | 19 | adantr 276 | . . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ ℝ ∧ 𝑎 ∈ 𝑈)) → ∀𝑞 ∈ ℝ ∀𝑟 ∈ ℝ (𝑞 < 𝑟 → (𝑞 ∈ 𝐿 ∨ 𝑟 ∈ 𝑈))) |
21 | simprr 531 | . . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ ℝ ∧ 𝑎 ∈ 𝑈)) → 𝑎 ∈ 𝑈) | |
22 | 7, 9, 11, 12, 14, 16, 18, 20, 21 | dedekindeulemuub 14366 | . . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ ℝ ∧ 𝑎 ∈ 𝑈)) → ∀𝑦 ∈ 𝐿 𝑦 < 𝑎) |
23 | brralrspcev 4073 | . . 3 ⊢ ((𝑎 ∈ ℝ ∧ ∀𝑦 ∈ 𝐿 𝑦 < 𝑎) → ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐿 𝑦 < 𝑥) | |
24 | 5, 22, 23 | syl2anc 411 | . 2 ⊢ ((𝜑 ∧ (𝑎 ∈ ℝ ∧ 𝑎 ∈ 𝑈)) → ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐿 𝑦 < 𝑥) |
25 | 4, 24 | rexlimddv 2609 | 1 ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐿 𝑦 < 𝑥) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 ∨ wo 709 = wceq 1363 ∈ wcel 2158 ∀wral 2465 ∃wrex 2466 ∩ cin 3140 ⊆ wss 3141 ∅c0 3434 class class class wbr 4015 ℝcr 7823 < clt 8005 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-13 2160 ax-14 2161 ax-ext 2169 ax-sep 4133 ax-pow 4186 ax-pr 4221 ax-un 4445 ax-setind 4548 ax-cnex 7915 ax-resscn 7916 ax-pre-ltwlin 7937 |
This theorem depends on definitions: df-bi 117 df-3an 981 df-tru 1366 df-fal 1369 df-nf 1471 df-sb 1773 df-eu 2039 df-mo 2040 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-ne 2358 df-nel 2453 df-ral 2470 df-rex 2471 df-rab 2474 df-v 2751 df-dif 3143 df-un 3145 df-in 3147 df-ss 3154 df-nul 3435 df-pw 3589 df-sn 3610 df-pr 3611 df-op 3613 df-uni 3822 df-br 4016 df-opab 4077 df-xp 4644 df-cnv 4646 df-pnf 8007 df-mnf 8008 df-xr 8009 df-ltxr 8010 df-le 8011 |
This theorem is referenced by: dedekindeulemlub 14369 |
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