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Mirrors > Home > ILE Home > Th. List > dedekindicclemub | GIF version |
Description: Lemma for dedekindicc 13744. The lower cut has an upper bound. (Contributed by Jim Kingdon, 15-Feb-2024.) |
Ref | Expression |
---|---|
dedekindicc.a | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
dedekindicc.b | ⊢ (𝜑 → 𝐵 ∈ ℝ) |
dedekindicc.lss | ⊢ (𝜑 → 𝐿 ⊆ (𝐴[,]𝐵)) |
dedekindicc.uss | ⊢ (𝜑 → 𝑈 ⊆ (𝐴[,]𝐵)) |
dedekindicc.lm | ⊢ (𝜑 → ∃𝑞 ∈ (𝐴[,]𝐵)𝑞 ∈ 𝐿) |
dedekindicc.um | ⊢ (𝜑 → ∃𝑟 ∈ (𝐴[,]𝐵)𝑟 ∈ 𝑈) |
dedekindicc.lr | ⊢ (𝜑 → ∀𝑞 ∈ (𝐴[,]𝐵)(𝑞 ∈ 𝐿 ↔ ∃𝑟 ∈ 𝐿 𝑞 < 𝑟)) |
dedekindicc.ur | ⊢ (𝜑 → ∀𝑟 ∈ (𝐴[,]𝐵)(𝑟 ∈ 𝑈 ↔ ∃𝑞 ∈ 𝑈 𝑞 < 𝑟)) |
dedekindicc.disj | ⊢ (𝜑 → (𝐿 ∩ 𝑈) = ∅) |
dedekindicc.loc | ⊢ (𝜑 → ∀𝑞 ∈ (𝐴[,]𝐵)∀𝑟 ∈ (𝐴[,]𝐵)(𝑞 < 𝑟 → (𝑞 ∈ 𝐿 ∨ 𝑟 ∈ 𝑈))) |
Ref | Expression |
---|---|
dedekindicclemub | ⊢ (𝜑 → ∃𝑥 ∈ (𝐴[,]𝐵)∀𝑦 ∈ 𝐿 𝑦 < 𝑥) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dedekindicc.um | . . 3 ⊢ (𝜑 → ∃𝑟 ∈ (𝐴[,]𝐵)𝑟 ∈ 𝑈) | |
2 | eleq1w 2238 | . . . 4 ⊢ (𝑟 = 𝑎 → (𝑟 ∈ 𝑈 ↔ 𝑎 ∈ 𝑈)) | |
3 | 2 | cbvrexv 2704 | . . 3 ⊢ (∃𝑟 ∈ (𝐴[,]𝐵)𝑟 ∈ 𝑈 ↔ ∃𝑎 ∈ (𝐴[,]𝐵)𝑎 ∈ 𝑈) |
4 | 1, 3 | sylib 122 | . 2 ⊢ (𝜑 → ∃𝑎 ∈ (𝐴[,]𝐵)𝑎 ∈ 𝑈) |
5 | simprl 529 | . . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ (𝐴[,]𝐵) ∧ 𝑎 ∈ 𝑈)) → 𝑎 ∈ (𝐴[,]𝐵)) | |
6 | dedekindicc.a | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
7 | 6 | adantr 276 | . . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ (𝐴[,]𝐵) ∧ 𝑎 ∈ 𝑈)) → 𝐴 ∈ ℝ) |
8 | dedekindicc.b | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ ℝ) | |
9 | 8 | adantr 276 | . . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ (𝐴[,]𝐵) ∧ 𝑎 ∈ 𝑈)) → 𝐵 ∈ ℝ) |
10 | dedekindicc.lss | . . . . 5 ⊢ (𝜑 → 𝐿 ⊆ (𝐴[,]𝐵)) | |
11 | 10 | adantr 276 | . . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ (𝐴[,]𝐵) ∧ 𝑎 ∈ 𝑈)) → 𝐿 ⊆ (𝐴[,]𝐵)) |
12 | dedekindicc.uss | . . . . 5 ⊢ (𝜑 → 𝑈 ⊆ (𝐴[,]𝐵)) | |
13 | 12 | adantr 276 | . . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ (𝐴[,]𝐵) ∧ 𝑎 ∈ 𝑈)) → 𝑈 ⊆ (𝐴[,]𝐵)) |
14 | dedekindicc.lm | . . . . 5 ⊢ (𝜑 → ∃𝑞 ∈ (𝐴[,]𝐵)𝑞 ∈ 𝐿) | |
15 | 14 | adantr 276 | . . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ (𝐴[,]𝐵) ∧ 𝑎 ∈ 𝑈)) → ∃𝑞 ∈ (𝐴[,]𝐵)𝑞 ∈ 𝐿) |
16 | 1 | adantr 276 | . . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ (𝐴[,]𝐵) ∧ 𝑎 ∈ 𝑈)) → ∃𝑟 ∈ (𝐴[,]𝐵)𝑟 ∈ 𝑈) |
17 | dedekindicc.lr | . . . . 5 ⊢ (𝜑 → ∀𝑞 ∈ (𝐴[,]𝐵)(𝑞 ∈ 𝐿 ↔ ∃𝑟 ∈ 𝐿 𝑞 < 𝑟)) | |
18 | 17 | adantr 276 | . . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ (𝐴[,]𝐵) ∧ 𝑎 ∈ 𝑈)) → ∀𝑞 ∈ (𝐴[,]𝐵)(𝑞 ∈ 𝐿 ↔ ∃𝑟 ∈ 𝐿 𝑞 < 𝑟)) |
19 | dedekindicc.ur | . . . . 5 ⊢ (𝜑 → ∀𝑟 ∈ (𝐴[,]𝐵)(𝑟 ∈ 𝑈 ↔ ∃𝑞 ∈ 𝑈 𝑞 < 𝑟)) | |
20 | 19 | adantr 276 | . . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ (𝐴[,]𝐵) ∧ 𝑎 ∈ 𝑈)) → ∀𝑟 ∈ (𝐴[,]𝐵)(𝑟 ∈ 𝑈 ↔ ∃𝑞 ∈ 𝑈 𝑞 < 𝑟)) |
21 | dedekindicc.disj | . . . . 5 ⊢ (𝜑 → (𝐿 ∩ 𝑈) = ∅) | |
22 | 21 | adantr 276 | . . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ (𝐴[,]𝐵) ∧ 𝑎 ∈ 𝑈)) → (𝐿 ∩ 𝑈) = ∅) |
23 | dedekindicc.loc | . . . . 5 ⊢ (𝜑 → ∀𝑞 ∈ (𝐴[,]𝐵)∀𝑟 ∈ (𝐴[,]𝐵)(𝑞 < 𝑟 → (𝑞 ∈ 𝐿 ∨ 𝑟 ∈ 𝑈))) | |
24 | 23 | adantr 276 | . . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ (𝐴[,]𝐵) ∧ 𝑎 ∈ 𝑈)) → ∀𝑞 ∈ (𝐴[,]𝐵)∀𝑟 ∈ (𝐴[,]𝐵)(𝑞 < 𝑟 → (𝑞 ∈ 𝐿 ∨ 𝑟 ∈ 𝑈))) |
25 | simprr 531 | . . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ (𝐴[,]𝐵) ∧ 𝑎 ∈ 𝑈)) → 𝑎 ∈ 𝑈) | |
26 | 7, 9, 11, 13, 15, 16, 18, 20, 22, 24, 25 | dedekindicclemuub 13737 | . . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ (𝐴[,]𝐵) ∧ 𝑎 ∈ 𝑈)) → ∀𝑦 ∈ 𝐿 𝑦 < 𝑎) |
27 | brralrspcev 4058 | . . 3 ⊢ ((𝑎 ∈ (𝐴[,]𝐵) ∧ ∀𝑦 ∈ 𝐿 𝑦 < 𝑎) → ∃𝑥 ∈ (𝐴[,]𝐵)∀𝑦 ∈ 𝐿 𝑦 < 𝑥) | |
28 | 5, 26, 27 | syl2anc 411 | . 2 ⊢ ((𝜑 ∧ (𝑎 ∈ (𝐴[,]𝐵) ∧ 𝑎 ∈ 𝑈)) → ∃𝑥 ∈ (𝐴[,]𝐵)∀𝑦 ∈ 𝐿 𝑦 < 𝑥) |
29 | 4, 28 | rexlimddv 2599 | 1 ⊢ (𝜑 → ∃𝑥 ∈ (𝐴[,]𝐵)∀𝑦 ∈ 𝐿 𝑦 < 𝑥) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 ∨ wo 708 = wceq 1353 ∈ wcel 2148 ∀wral 2455 ∃wrex 2456 ∩ cin 3128 ⊆ wss 3129 ∅c0 3422 class class class wbr 4000 (class class class)co 5868 ℝcr 7788 < clt 7969 [,]cicc 9865 |
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 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4118 ax-pow 4171 ax-pr 4205 ax-un 4429 ax-setind 4532 ax-cnex 7880 ax-resscn 7881 ax-pre-ltirr 7901 ax-pre-ltwlin 7902 ax-pre-lttrn 7903 |
This theorem depends on definitions: df-bi 117 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-rab 2464 df-v 2739 df-sbc 2963 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-br 4001 df-opab 4062 df-id 4289 df-po 4292 df-iso 4293 df-xp 4628 df-rel 4629 df-cnv 4630 df-co 4631 df-dm 4632 df-iota 5173 df-fun 5213 df-fv 5219 df-ov 5871 df-oprab 5872 df-mpo 5873 df-pnf 7971 df-mnf 7972 df-xr 7973 df-ltxr 7974 df-le 7975 df-icc 9869 |
This theorem is referenced by: (None) |
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