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Mirrors > Home > ILE Home > Th. List > dedekindicclemeu | GIF version |
Description: Lemma for dedekindicc 12819. Part of proving uniqueness. (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))) |
dedekindicc.ab | ⊢ (𝜑 → 𝐴 < 𝐵) |
dedekindicclemeu.are | ⊢ (𝜑 → 𝐶 ∈ (𝐴[,]𝐵)) |
dedekindicclemeu.ac | ⊢ (𝜑 → (∀𝑞 ∈ 𝐿 𝑞 < 𝐶 ∧ ∀𝑟 ∈ 𝑈 𝐶 < 𝑟)) |
dedekindicclemeu.bre | ⊢ (𝜑 → 𝐷 ∈ (𝐴[,]𝐵)) |
dedekindicclemeu.bc | ⊢ (𝜑 → (∀𝑞 ∈ 𝐿 𝑞 < 𝐷 ∧ ∀𝑟 ∈ 𝑈 𝐷 < 𝑟)) |
dedekindicclemeu.lt | ⊢ (𝜑 → 𝐶 < 𝐷) |
Ref | Expression |
---|---|
dedekindicclemeu | ⊢ (𝜑 → ⊥) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | breq1 3940 | . . . 4 ⊢ (𝑞 = 𝐶 → (𝑞 < 𝐶 ↔ 𝐶 < 𝐶)) | |
2 | dedekindicclemeu.ac | . . . . . 6 ⊢ (𝜑 → (∀𝑞 ∈ 𝐿 𝑞 < 𝐶 ∧ ∀𝑟 ∈ 𝑈 𝐶 < 𝑟)) | |
3 | 2 | simpld 111 | . . . . 5 ⊢ (𝜑 → ∀𝑞 ∈ 𝐿 𝑞 < 𝐶) |
4 | 3 | adantr 274 | . . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝐿) → ∀𝑞 ∈ 𝐿 𝑞 < 𝐶) |
5 | simpr 109 | . . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝐿) → 𝐶 ∈ 𝐿) | |
6 | 1, 4, 5 | rspcdva 2798 | . . 3 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝐿) → 𝐶 < 𝐶) |
7 | dedekindicc.a | . . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
8 | dedekindicc.b | . . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ ℝ) | |
9 | iccssre 9768 | . . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ⊆ ℝ) | |
10 | 7, 8, 9 | syl2anc 409 | . . . . . 6 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℝ) |
11 | dedekindicclemeu.are | . . . . . 6 ⊢ (𝜑 → 𝐶 ∈ (𝐴[,]𝐵)) | |
12 | 10, 11 | sseldd 3103 | . . . . 5 ⊢ (𝜑 → 𝐶 ∈ ℝ) |
13 | 12 | ltnrd 7899 | . . . 4 ⊢ (𝜑 → ¬ 𝐶 < 𝐶) |
14 | 13 | adantr 274 | . . 3 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝐿) → ¬ 𝐶 < 𝐶) |
15 | 6, 14 | pm2.21fal 1352 | . 2 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝐿) → ⊥) |
16 | breq2 3941 | . . . 4 ⊢ (𝑟 = 𝐷 → (𝐷 < 𝑟 ↔ 𝐷 < 𝐷)) | |
17 | dedekindicclemeu.bc | . . . . . 6 ⊢ (𝜑 → (∀𝑞 ∈ 𝐿 𝑞 < 𝐷 ∧ ∀𝑟 ∈ 𝑈 𝐷 < 𝑟)) | |
18 | 17 | simprd 113 | . . . . 5 ⊢ (𝜑 → ∀𝑟 ∈ 𝑈 𝐷 < 𝑟) |
19 | 18 | adantr 274 | . . . 4 ⊢ ((𝜑 ∧ 𝐷 ∈ 𝑈) → ∀𝑟 ∈ 𝑈 𝐷 < 𝑟) |
20 | simpr 109 | . . . 4 ⊢ ((𝜑 ∧ 𝐷 ∈ 𝑈) → 𝐷 ∈ 𝑈) | |
21 | 16, 19, 20 | rspcdva 2798 | . . 3 ⊢ ((𝜑 ∧ 𝐷 ∈ 𝑈) → 𝐷 < 𝐷) |
22 | dedekindicclemeu.bre | . . . . . 6 ⊢ (𝜑 → 𝐷 ∈ (𝐴[,]𝐵)) | |
23 | 10, 22 | sseldd 3103 | . . . . 5 ⊢ (𝜑 → 𝐷 ∈ ℝ) |
24 | 23 | ltnrd 7899 | . . . 4 ⊢ (𝜑 → ¬ 𝐷 < 𝐷) |
25 | 24 | adantr 274 | . . 3 ⊢ ((𝜑 ∧ 𝐷 ∈ 𝑈) → ¬ 𝐷 < 𝐷) |
26 | 21, 25 | pm2.21fal 1352 | . 2 ⊢ ((𝜑 ∧ 𝐷 ∈ 𝑈) → ⊥) |
27 | dedekindicclemeu.lt | . . 3 ⊢ (𝜑 → 𝐶 < 𝐷) | |
28 | breq2 3941 | . . . . 5 ⊢ (𝑟 = 𝐷 → (𝐶 < 𝑟 ↔ 𝐶 < 𝐷)) | |
29 | eleq1 2203 | . . . . . 6 ⊢ (𝑟 = 𝐷 → (𝑟 ∈ 𝑈 ↔ 𝐷 ∈ 𝑈)) | |
30 | 29 | orbi2d 780 | . . . . 5 ⊢ (𝑟 = 𝐷 → ((𝐶 ∈ 𝐿 ∨ 𝑟 ∈ 𝑈) ↔ (𝐶 ∈ 𝐿 ∨ 𝐷 ∈ 𝑈))) |
31 | 28, 30 | imbi12d 233 | . . . 4 ⊢ (𝑟 = 𝐷 → ((𝐶 < 𝑟 → (𝐶 ∈ 𝐿 ∨ 𝑟 ∈ 𝑈)) ↔ (𝐶 < 𝐷 → (𝐶 ∈ 𝐿 ∨ 𝐷 ∈ 𝑈)))) |
32 | breq1 3940 | . . . . . . 7 ⊢ (𝑞 = 𝐶 → (𝑞 < 𝑟 ↔ 𝐶 < 𝑟)) | |
33 | eleq1 2203 | . . . . . . . 8 ⊢ (𝑞 = 𝐶 → (𝑞 ∈ 𝐿 ↔ 𝐶 ∈ 𝐿)) | |
34 | 33 | orbi1d 781 | . . . . . . 7 ⊢ (𝑞 = 𝐶 → ((𝑞 ∈ 𝐿 ∨ 𝑟 ∈ 𝑈) ↔ (𝐶 ∈ 𝐿 ∨ 𝑟 ∈ 𝑈))) |
35 | 32, 34 | imbi12d 233 | . . . . . 6 ⊢ (𝑞 = 𝐶 → ((𝑞 < 𝑟 → (𝑞 ∈ 𝐿 ∨ 𝑟 ∈ 𝑈)) ↔ (𝐶 < 𝑟 → (𝐶 ∈ 𝐿 ∨ 𝑟 ∈ 𝑈)))) |
36 | 35 | ralbidv 2438 | . . . . 5 ⊢ (𝑞 = 𝐶 → (∀𝑟 ∈ (𝐴[,]𝐵)(𝑞 < 𝑟 → (𝑞 ∈ 𝐿 ∨ 𝑟 ∈ 𝑈)) ↔ ∀𝑟 ∈ (𝐴[,]𝐵)(𝐶 < 𝑟 → (𝐶 ∈ 𝐿 ∨ 𝑟 ∈ 𝑈)))) |
37 | dedekindicc.loc | . . . . 5 ⊢ (𝜑 → ∀𝑞 ∈ (𝐴[,]𝐵)∀𝑟 ∈ (𝐴[,]𝐵)(𝑞 < 𝑟 → (𝑞 ∈ 𝐿 ∨ 𝑟 ∈ 𝑈))) | |
38 | 36, 37, 11 | rspcdva 2798 | . . . 4 ⊢ (𝜑 → ∀𝑟 ∈ (𝐴[,]𝐵)(𝐶 < 𝑟 → (𝐶 ∈ 𝐿 ∨ 𝑟 ∈ 𝑈))) |
39 | 31, 38, 22 | rspcdva 2798 | . . 3 ⊢ (𝜑 → (𝐶 < 𝐷 → (𝐶 ∈ 𝐿 ∨ 𝐷 ∈ 𝑈))) |
40 | 27, 39 | mpd 13 | . 2 ⊢ (𝜑 → (𝐶 ∈ 𝐿 ∨ 𝐷 ∈ 𝑈)) |
41 | 15, 26, 40 | mpjaodan 788 | 1 ⊢ (𝜑 → ⊥) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 103 ↔ wb 104 ∨ wo 698 = wceq 1332 ⊥wfal 1337 ∈ wcel 1481 ∀wral 2417 ∃wrex 2418 ∩ cin 3075 ⊆ wss 3076 ∅c0 3368 class class class wbr 3937 (class class class)co 5782 ℝcr 7643 < clt 7824 [,]cicc 9704 |
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 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-setind 4460 ax-cnex 7735 ax-resscn 7736 ax-pre-ltirr 7756 ax-pre-ltwlin 7757 ax-pre-lttrn 7758 |
This theorem depends on definitions: df-bi 116 df-3or 964 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-nel 2405 df-ral 2422 df-rex 2423 df-rab 2426 df-v 2691 df-sbc 2914 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-br 3938 df-opab 3998 df-id 4223 df-po 4226 df-iso 4227 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-iota 5096 df-fun 5133 df-fv 5139 df-ov 5785 df-oprab 5786 df-mpo 5787 df-pnf 7826 df-mnf 7827 df-xr 7828 df-ltxr 7829 df-le 7830 df-icc 9708 |
This theorem is referenced by: dedekindicclemicc 12818 |
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